Negative Index Photonic Crystals Superlattices and Zero Phase Delay Lines

The second volume of the book concerns the characterization approach of photonic crystals, photonic crystal lasers, photonic crystal waveguides and plasmonics including the introduction of innovative systems and materials. Photonic crystal materials promises to enable all-optical computer circuits and could also be used to make ultra low-power light sources. Researchers have studied lasers from microscopic cavities in photonic crystals that act as reflectors to intensify the collisions between photons and atoms that lead to lazing, but these lasers have been optically-pumped, meaning they are driven by other lasers. Moreover, the physical principles behind the phenomenon of slow light in photonic crystal waveguides, as well as their practical limitations, are discussed. This includes the nature of slow light propagation, its bandwidth limitation, coupling of modes and particular kind terminating photonic crystals with metal surfaces allowing to propagate in surface plasmon-polariton waves. The goal of the second volume is to provide an overview about the listed issues.

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, 328 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 zeron , or zero-order, bandgap [30,34]. 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, www.intechopen.com 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 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 zeron 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 zeron gap frequencies, optical phase, and the effective refractive indices agree well between the series of measurements and the complete theoretical analysis and simulations.

Theory
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 zeron 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/2c, or 1520 to 1560 nm wavelengths, such as reported earlier for near-field imaging [22].
The zeron 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 m) 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 zeron gap edge (1560 nm). The period of the superlattice is equal to www.intechopen.com Photonic Crystals -Innovative Systems, Lasers and Waveguides 330 =d 1 + d 2 where d 1 (2) is the thickness of the PhC (PIM) layer in the primary unit cell. Since a zeron 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]. 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]    holds. This relation implies that the dispersion relation has no real solution for  unless 11 n 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 , 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.

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 r 2 to 5/6 of the original value of the radius r 1 (r 2 /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 www.intechopen.com Photonic Crystals -Innovative Systems, Lasers and Waveguides 332 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.

Spatial field distribution
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 interference output has 14dB difference.

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.

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  Figure 1d is determined from the relation k = ω|n|/c (note  Figure 6b. The grid size resolution in all our numerical simulations is 0.0833a (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  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 k o Λ/π for the two deigns are summarized in Table 1 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 zeron gap location has not changed and, as expected, it becomes deeper as the number of stacks increases.

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.

Device nanofabrication and experiments
Our theoretical predictions are validated by a series of experiments. Thus, we have fabricated in a single-crystal silicon-on-insulator substrate samples with 3, 5 and 8 stacks whose PhC layers have thickness of 1 d 3 . 5 3 a  . The silicon device height is 320 nm and the silicon oxide cladding thickness is 1 m. The PhC superlattice is lithographically patterned with a 248-nm lithography scanner, and the Si is plasma-etched. Figure 1b shows an example of a fabricated photonic crystal superlattice with 8 stacks. The fabrication disorder in the PhC slab was statistically parameterized [37], with resulting hole radius 122.207  1.207 nm, lattice period 421.78  1.26 nm (~ 0:003a), and ellipticity 1.21 nm  0.56 nm. These small variations are below ~0:05a disorder theoretical target [46]. Incident light from tunable lasers between 1480 nm to 1690 nm (0.248 to 0.284 in normalized frequency of a/2c) is coupled into the chip via tapered lensed fibers with manual fiber polarization control. The transmission for the TM polarization is measured, with each transmission measurement averaged over three scans. Figure 8a shows the transmission spectrum for design 1 (with d 2 /d 1 = 0.746); it shows two distinct spectral dips, centered at 1520 nm (a/2c  0.276) and 1585 nm (a/2c  0.265). We then repeat these transmission measurements for a second design (design 2; d 2 /d 1 = 0.794); the corresponding results are shown in Figure 8b. Similar to the spectra in Figure 8a, this figure shows a distinct spectral dip, located near the normalized frequency a/2c  0.272, i.e. at =1543 nm. Furthermore, the frequency spectral dip at a/2c  0.262 is weaker than in design 1, which is due to the fact that below a certain frequency, a/2c  0.265, the detected power is not high enough for observing the spectral features. Figure 8c shows the near-infrared image captured with incident laser at 1550 nm and corresponds to the design 1. The spatially alternating vertical stripes show radiation scattered at the interfaces between the PhC and the homogeneous layers and confirm the transmission of the light through the superlattice. The near-infrared images also confirm the existence of the dip in the transmission spectrum, with most of the light being reflected and only a small amount propagating out of the output facet of the third stack (in this figure , the incident laser is tuned to the zeron gap frequency).
To better understand the results of these measurements, we repeated the FDTD simulations for the case of the fabricated devices. A good match between the results of the measurements and those of simulations, in terms of absolute values of the frequencies, has been observed for both values of the ratio d 2 /d 1 (design 1 and 2) and varying stack numbers (design 1). The theoretical predictions are shown as the dotted lines in Figure 8a    It has been pointed out that zeron gaps can be omnidirectional [28,41]; however, in our case, due to the anisotropy of the index of refraction of the PhC, the zeron 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 zeron gap can be easily tuned as we show in the next section. Importantly, we note the demonstration of these zeron gap structures can have potential applications as delay lines with zero phase differences which we also show later in this chapter.

Tunability of zero-n gap
Next, in order to demonstrate the tunable character of the zero-n bandgaps, we performed transmission experiments on four sets of binary superlattices, with each set having different superlattice ratios: d 2 /d 1 =0.74 (Figure 9a), d 2 /d 1 =0.76 (Figure 9b), d 2 /d 1 =0.78 (Figure 9d), and d 2 /d 1 =0.8 (Figure 9e). In all our experiments the negative index PhC has the same parameters as those given above. Each set has three devices of different periods , with the negative index PhC layer in the superlattice spanning 7, 9, and 11 unit cells along the z-axis similar to the numerical study in 3.1, so that the thickness of this layer is 1 d 3 .  Table 2). Here, n 1 and n 2 are the effective mode indices in the PhC and homogeneous layers respectively at the corresponding wavelengths. For the three devices in each set, we designed 7 super-periods (SPs) for the devices with 7 unit cells of PhC and 5 SPs for those with 9 and 11 unit cells of PhC (these designs ensure a sufficient signal-to-noise ratio for the transmission measurements). In these experiments we have tested both the existence of the zero-n bandgap as well as its tunability. For the three devices belonging to each set, we observed the zero-n bandgap at the same frequency whereas the spectral locations of the other bandgaps were observed to shift with the frequency -this confirms the zero-n bandgap does not depend on the total superperiod length gap existence dependent only    Table 2. Calculated parameters of the devices in the Figure 9 (units in m).
www.intechopen.com on the condition of path-averaged zero index: n 1 d 1 + n 2 d 2 = 0) while the frequency of the regular 1D PhC Bragg bandgaps does depend on . Our measurements show that the invariant, zeron , bandgap is located at 1525.5 nm, 1535.2 nm, 1546.3 nm, and 1556.5 nm, respectively (averaged over the three devices in each set). The slight red-shift with increasing number of unit cells in each set is due to effects of edge termination between the PhC and the homogeneous slab. Furthermore, when we tuned the ratio d 2 /d 1 and repeated these same experiments we observed a redshift of the zero-n mid-gap frequency as we increased the ratio d 2 /d 1 . This result is explained by the fact that for the negative index band the refractive index of the 2D hexagonal PhC decreases with respect to the wavelength (see Figure 3c) and therefore when the length of the PIM layer in the 1D binary superlattice increases (higher d 2 /d 1 ), the wavelength at which the effective index cancels is red-shifted. The effective index of the PIM www.intechopen.com Photonic Crystals -Innovative Systems, Lasers and Waveguides 342 layer, n 2 , is calculated numerically and for the asymmetric TM slab waveguide mode corresponds to, for example, 2.648 at 1550 nm. By using these n 1 (Figure 3c) and n 2 values, we determined the average refractive index for the different d 2 /d 1 ratios as summarized in Figure 9f. A distinctive red-shift in the zero-n gap location is observed with increasing d 2 /d 1 ratios from the numerically modeling, demonstrating good agreement with the experimental measurements (Figure 9a, 9b, 9d, and 9e) without any parameter fitting in the analysis. Furthermore, Figure 9c shows how the spectral features of the zero-n b a n d g a p changes with increasing the number of superperiods and the results are similar to those in Figure8d as expected. We note that this is the first rigorous and complete experimental confirmation of invariant and tunable character of zero-n bandgaps in photonic superlattices containing negative index PhCs.

Zero phase delay lines
Next, we prove that the total phase accumulation in the superlattice is zero. For this, we performed phase measurements for the designs with three different sets of measurements: In these series of measurements we used a single mode channel waveguide for the reference arm of the MZI -this enables a series of interference fringes at the output, which can be used to determine the phase change by analyzing the spectral location of the fringes and their free spectral range (FSR). In most free-space interferometric applications, the phase difference leading to interference originates from the physical length difference between the two arms, but in integrated photonic circuits this delay can easily be modulated by the imbalance in the refractive indices of two arms [47]. For (a) and (c), we examined three devices, namely, superlattices with 5, 6, and 7 SPs and for (b) we tested superlattices with 5 and 7 SPs. When we designed these devices, we modified the MZI such that when we added a SP to the superlattice the length of the adiabatic transition arms was carefully increased by /2, making the horizontal single mode channel waveguides shorter (from L 2 to L 2 -/2, at both sides in Figure 1a). This change is compensated by adding the same length to the vertical part (from L 3 to L 3 +/2 on both sides in Figure 1a). As a result, the only phase difference between devices is due to the additional SPs. This procedure is explained in Section 5.1 in detail. Figure 10 shows a schematic representation of a device with 2 superperiods and the integrated Mach Zehnder Interferometer is modified after introducing the third superperiod. The adiabatic region remains unchanged if L 1 is increased to L 1 +Λ and L 2 is shortened by Λ/2, in both the input and output sides of the device. To keep the total length of the waveguide unchanged, the length L 3 is increased to L 3 + Λ/2. This procedure is used each time a superperiod is added to the structure. In addition, to be able to compare devices with different number of unit cells in the PhC layer, a common reference point is used for all devices that have the same d 2 /d 1 ratio.

Device modification for phase measurements
In our implementation, the interferometer output intensity is given as: 12  where n wg , n slab_1 , n sl are the effective mode refractive indices of the channel waveguide, the adiabatic slab in arm 1, and the zero-index superlattice, respectively. L i denotes the corresponding lengths. Fig. 10. Schematic representation of the device modification induced by adding a superperiod. a, Integrated MZI of a device with 2 superperiods and a channel waveguide. b, Device modifications after the third superperiod is added. The length of the channel waveguide remains the same so as the effect of the additional superperiod is isolated.
We note that the difference between the physical path length of the channel waveguides on both arms is designed to be equal to the physical path length of the tapering slab. Thus we have:  Figure 11a shows the interference pattern for d 2 /d 1 =0.78 with 7 unit cells in the PhC layer.

Experimental results for zero phase
As can be seen in this figure, outside the zero-n spectral region the fringes differ from each other both in wavelength and the FSR, but overlap almost perfectly within the zero-n spectral domain.
To illustrate the phase evolution, we show in Figure 12a the FSR values for each of the devices examined -specifically we calculate the spectral spacing between the transmission minima and plot its dependence on the center wavelength between the two neighboring minima. As these measurements illustrate, in the zeron spectral domain the FSR corresponding to each of the devices approaches the same value, indicating that the corresponding phase difference is zero or, alternatively, that the optical path remains unchanged. This is a surprising conclusion since the physical path is certainly not the same in all the cases. This apparent paradox has a simple explanation: although the physical path varies among the three cases the optical path is the same (and equal to zero) as the spatially averaged refractive index of the three superlattices vanishes. In other words, within the zeron spectral region the photonic superlattice emulates the properties of a zero phase delay line. The output corresponding to the structures with d 2 /d 1 =0.8 and 7 unit cells in the PhC layer is shown in Figure 11b whereas the FSR values are plotted in Figure 12b. Finally, Figure 11c and Figure 12c show the interference patterns for the case of d 2 /d 1 =0.8 and 9 unit cells in the PhC layer. Again, both the FSR (Figure 12b-c) and the absolute wavelength values (Figure 11b-c) overlap, proving the zero phase variation across the superlattice.
It should be noted that in all our plots of experimental data we have used the raw data and as such there is no data post-processing, except for the intensity rescaling. Measurements are taken 3 times with 500 pm resolution for Figure 3d and  Figure 11e and Figure 11b-c in terms of the center frequency of the zeron region. This is so because of the fabrication differences between the samples. For Figure  11b-c, the r/a ratio was ~5% smaller (~0.264) resulting in the shift of the band structure to lower frequencies, and, consequently to a shift of the zero-n b a n d g a p . W e v e r i f i e d t h e location of the zero-n bandgap (~1565 nm) by performing the transmission measurements described before. Thus, the spectral location of the zero-n bandgap can be tracked from the phase measurements, as the spectral region of small amplitude oscillations in the transmission spectra correspond to the zero-n b a n d g a p s . In summary, we have demonstrated for the first time zero-phase delay in negative-positive index superlattices, in addition to the simultaneous observations of deterministic zero-n gaps that can remain invariant to geometric changes and band-to-band transitions in negative-positive index photonic crystal superlattices. Through the interferometric measurements, the transmissive binary superlattices with varying lengths are shown unequivocally to enable the absolute control of the optical phase. The engineered control of the phase delay in these near-zero superlattices can be implemented in chip-scale www.intechopen.com transmission lines with deterministic phase array control, even with technological potential in phase-insensitive image processing, phase-invariant field for electromagnetic cloaking, and the arbitrary radiation wavefront reshaping of antennae from first principles. Fig. 12. Free spectral range wavelength dependence corresponding to superlattices in Figure 4. a-c, Free spectral range extracted from the data in Figure 4a-c. At the zero-n bandgap wavelength, the free spectral range converges to the same value, which proves the zero phase contribution from the added superperiods.