of unit cells *d1 d2 Λ* 7 2.56 2.05 4.62 9 3.30 2.64 5.93 11 4.03 3.22 7.25

Table 2. **Calculated parameters of the devices in the Figure 9 (units in m).** 

gap existence dependent only

, with the negative

as those given above. Each set has three devices of different periods

bandgap does not depend on the total superperiod length

**4. Tunability of zero-n gap** 

[ \_

Figure 9a

Figure 9b

Figure 9d

Figure 9e

on the condition of path-averaged zero index: *n1d1* + *n2d2* = 0) while the frequency of the regular 1D PhC Bragg bandgaps does depend on . Our measurements show that the invariant, zero- n , 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.

Fig. 9. **Experimental verification of period-invariance and tunability of zero-** n **bandgaps. a,** Experimental verification of the zero- n bandgap in superlattices with varying period (). The ratio *d*2/*d*1 =0.74 and =4.46 μm for black (solid), 5.74 μm for red (dashed), and 7.01 μm for green (dotted) curves (*a.u.* arbitrary units). The lightly shaded regions in all panels denote the negative index regions. **b,** Same as in **a**, but for *d*2/*d*1 =0.76. =4.51 μm for black (solid), 5.80 μm for red (dashed), and 7.09 μm for green (dotted) curves. **c,** Transmission spectra for superlattices with *d*2/*d*1 =0.76, containing 5, 6, and 7 unit cells (UC). Each PhC layer contains 7 unit cells (*d1*=2.564 μm *,* =4.51μm). **d,** Same as in **a**, but for *d*2/*d*1 =0.78. =4.56 μm for black (solid), 5.87μm for red (dashed), and 7.17 μm for green (dotted) curves. **e,** Same as in **a**, but for *d*2/*d*1 =0.80. =4.61 μm for black (solid), 5.93 μm for red (dashed), and 7.25 μm for green (dotted) curves. **f,** Calculated effective index of refraction for the superlattices with the ratios in **a**, **b**, **d**, and **e**. The wavelengths at which the average index of refraction cancels agree very well with the measured values.

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

Negative Index Photonic Crystals Superlattices and Zero Phase Delay Lines 343

arms (denoted by subscript 1 and 2). Considering our implementation, the phase

 

is the phase difference (or imbalance) between the modes propagating in the two

<sup>2</sup> n L n L nL n L

 

 

12 wg wg \_ 1 slab \_ 1 slab \_ 1 sl sl wg wg \_ 2

wg wg \_ 1 wg \_ 2 slab \_ 1 slab \_ 1 sl sl

where *nwg*, *nslab\_1*, *nsl* are the effective mode refractive indices of the channel waveguide, the adiabatic slab in arm 1, and the zero-index superlattice, respectively. *Li* denotes the

2 2 n L L n L nL

Fig. 10. **Schematic representation of the device modification induced by adding a** 

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.

**superperiod. a,** Integrated MZI of a device with 2 superperiods and a channel waveguide. **b,**

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

can be

(4)

where 

have:

decomposed as:

corresponding lengths.

layer, *n2*, is calculated numerically and for the asymmetric TM slab waveguide mode corresponds to, for example, 2.648 at 1550 nm. By using these *n1* (Figure 3c) and *n2* 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 bandgap 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.

### **5. 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: **(a)** *d*2/*d*1=0.78 and 7 unit cells in the PhC layer; **(b)** *d*2/*d*1= 0.8 and 7 unit cells in the PhC layer; and **(c)** *d*2/*d*1= 0.8 and 9 unit cells in the PhC layer. 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 *L2* to *L2*-/2, at both sides in Figure 1a). This change is compensated by adding the same length to the vertical part (from *L3* to *L3*+/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.

#### **5.1 Device modification for phase measurements**

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 *L1* is increased to *L1*+*Λ* and *L2* 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 *L3* is increased to *L3+ 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 *d2/d1* ratio.

In our implementation, the interferometer output intensity is given as:

$$\mathbf{I} = \mathbf{I}\_1 + \mathbf{I}\_2 + 2\sqrt{\mathbf{I}\_1 \mathbf{I}\_2} \cos \phi \tag{3}$$

layer, *n2*, is calculated numerically and for the asymmetric TM slab waveguide mode corresponds to, for example, 2.648 at 1550 nm. By using these *n1* (Figure 3c) and *n2* 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 bandgap 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

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: **(a)** *d*2/*d*1=0.78 and 7 unit cells in the PhC layer; **(b)** *d*2/*d*1= 0.8 and 7 unit cells in the PhC layer; and **(c)** *d*2/*d*1= 0.8 and 9 unit cells in the PhC layer. 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

Figure 1a). This change is compensated by adding the same length to the vertical part (from

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 *L1* is increased to *L1*+*Λ* and *L2* 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 *L3* is increased to *L3+ 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

1 2 12 I I I 2 I I cos

devices is due to the additional SPs. This procedure is explained in Section 5.1 in detail.

/2 on both sides in Figure 1a). As a result, the only phase difference between

(3)

/2, making the

/2, at both sides in

length of the adiabatic transition arms was carefully increased by

horizontal single mode channel waveguides shorter (from *L2* to *L2*-

In our implementation, the interferometer output intensity is given as:

**5.1 Device modification for phase measurements** 

devices that have the same *d2/d1* ratio.

superlattices containing negative index PhCs.

**5. Zero phase delay lines** 

*L3* to *L3*+

where is the phase difference (or imbalance) between the modes propagating in the two arms (denoted by subscript 1 and 2). Considering our implementation, the phase can be decomposed as:

$$\begin{split} \phi = \phi\_1 - \phi\_2 &= \frac{2\pi}{\lambda} \left[ \left( \mathbf{n}\_{\text{wg}} \mathbf{L}\_{\text{wg}\_- - 1} + \mathbf{n}\_{\text{slab}\_- 1} \mathbf{L}\_{\text{slab}\_- 1} + \mathbf{n}\_{\text{sl}} \mathbf{L}\_{\text{sl}} \right) - \mathbf{n}\_{\text{wg}} \mathbf{L}\_{\text{wg}\_- 2} \right] \\ &= \frac{2\pi}{\lambda} \left[ \mathbf{n}\_{\text{wg}} \left( \mathbf{L}\_{\text{wg}\_- - 1} - \mathbf{L}\_{\text{wg}\_- 2} \right) + \mathbf{n}\_{\text{slab}\_- 1} \mathbf{L}\_{\text{slab}\_- - 1} \right] + \frac{2\pi}{\lambda} \mathbf{n}\_{\text{sl}} \mathbf{L}\_{\text{sl}} \end{split} \tag{4}$$

where *nwg*, *nslab\_1*, *nsl* are the effective mode refractive indices of the channel waveguide, the adiabatic slab in arm 1, and the zero-index superlattice, respectively. *Li* 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:

Negative Index Photonic Crystals Superlattices and Zero Phase Delay Lines 345

resolutions for Figure11a and 200 pm and 500 pm resolutions for Figure 11b-c. There is ~ 1% deviation between 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 bandgap. We verified the 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

Fig. 11. **Phase measurements. a,** Output of the MZI with increasing number of superperiods (5, 6, and 7 SP) on one arm and with length-adjusted single mode channel waveguides on the other arm. Each PhC layer contains 7 unit cells (*d1*=2.564 μm) and *d*2/*d*1 =0.78. **b,** Same as

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

in **a**, but for *d*2/*d*1 =0.8. **c,** Same as in b, with each PhC layer containing 9 unit cells

(*d1*=3.297 μm).

transmission spectra correspond to the zero- n bandgaps.

$$\begin{split} \cos[\phi] &= \cos\left[\frac{2\pi}{\lambda} \left( \left( -\mathbf{n}\_{\text{wg}} \mathbf{L}\_{\text{Slab}\_{-1}} + \mathbf{n}\_{\text{Slab}\_{-1}} \mathbf{L}\_{\text{Slab}\_{-1}} \right) + \mathbf{n}\_{\text{sl}} \mathbf{L}\_{\text{sl}} \right) \right] \\ &= \cos\left[\frac{2\pi}{\lambda} \left( \left( \mathbf{n}\_{\text{Slab}\_{-1}} - \mathbf{n}\_{\text{wg}} \right) \mathbf{L}\_{\text{Slab}\_{-1}} + \mathbf{n}\_{\text{sl}} \mathbf{L}\_{\text{sl}} \right) \right] \\ &= \cos\left[\phi\_{\text{slab}-\text{wg}} + \phi\_{\text{sl}} \right] \end{split} \tag{5}$$

The mode indices *nwg*, *nslab\_1*, *nsl* have different frequency dispersion. We kept the phase (*slab\_wg*), arising from slab \_ 1 wg slab \_ 1 <sup>2</sup> n nL , constant between different devices in each set of measurements by simply ensuring that the physical lengths and widths of the slabs are the same for each nanofabricated device. The remaining phase variation therefore is generated only by the photonic crystal superlattice (*sl* = sl sl <sup>2</sup> n L ). If *nsl* is equal to zero, *sl* is zero too, hence the total phase difference in the interferometer arises only from the *slab\_wg*

component and is the same for all the devices in each set. Therefore, the sinusoidal oscillations in the transmission and the free spectral range are determined only by *slab\_wg*.

#### **5.2 Experimental results for zero phase**

Figure 11a shows the interference pattern for *d*2/*d*1=0.78 with 7 unit cells in the PhC layer. 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 zero- n 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 zero- n 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 11a-e; 100 pm and 500 pm

wg Slab \_ 1 Slab \_ 1 Slab \_ 1 sl sl

, constant between different devices in each set

*sl* = sl sl <sup>2</sup> n L 

in the interferometer arises only from the

). If *nsl* is equal to zero,

(5)

*sl* is

*slab\_wg*

*slab\_wg*.

<sup>2</sup> cos n n L n L

 

 

slab wg sl

<sup>2</sup> n nL

cos

(*slab\_wg*), arising from slab \_ 1 wg slab \_ 1

zero too, hence the total phase difference

**5.2 Experimental results for zero phase** 

zero phase variation across the superlattice.

spectral domain.

generated only by the photonic crystal superlattice (

Slab \_ 1 wg Slab \_ 1 sl sl

The mode indices *nwg*, *nslab\_1*, *nsl* have different frequency dispersion. We kept the phase

of measurements by simply ensuring that the physical lengths and widths of the slabs are the same for each nanofabricated device. The remaining phase variation therefore is

component and is the same for all the devices in each set. Therefore, the sinusoidal

Figure 11a shows the interference pattern for *d*2/*d*1=0.78 with 7 unit cells in the PhC layer. 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

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 zero- n 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 zero- n 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

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 11a-e; 100 pm and 500 pm

oscillations in the transmission and the free spectral range are determined only by

<sup>2</sup> cos cos n L n L n L

resolutions for Figure11a and 200 pm and 500 pm resolutions for Figure 11b-c. There is ~ 1% deviation between 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 bandgap. We verified the 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 bandgaps.

Fig. 11. **Phase measurements. a,** Output of the MZI with increasing number of superperiods (5, 6, and 7 SP) on one arm and with length-adjusted single mode channel waveguides on the other arm. Each PhC layer contains 7 unit cells (*d1*=2.564 μm) and *d*2/*d*1 =0.78. **b,** Same as in **a**, but for *d*2/*d*1 =0.8. **c,** Same as in b, with each PhC layer containing 9 unit cells (*d1*=3.297 μm).

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

Negative Index Photonic Crystals Superlattices and Zero Phase Delay Lines 347

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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.

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