1. Introduction

Slowing down the propagation of light might enable a broad range of applications including delay lines, highly non-linear devices, all-optical buffer, and quantum computing [1–4]. To this end, dispersion engineering in photonic-crystal (PhC) waveguides is viewed as an important direction for research. In PhC waveguides, the slow-light frequency regime has been extensively studied to characterize group-index (n_g) and group-velocity dispersion (GVD) [5–6]. Furthermore, structural tuning of PhC waveguides in order to induce an inflection point on the guided-mode dispersion-band, i.e. dispersion compensation resulting in high n_g with relatively low GVD, has been a focus of various research groups [7–9].

Higher-order dispersion has been largely ignored owing to difficulties in simultaneous measurement of both transmission amplitude and phase in one set of experiments [5–9]. Dispersion parameters obtained from experiments relying on transmission measurement through PhC waveguides alone are somewhat ambiguous, because Fabry-Perot oscillations arising from the experimental set-up become superposed onto the transmission spectrum and cause loss of phase information [9]. Nevertheless, investigating the role of higher-order dispersion is important in order to understand pulse broadening due to chromatic dispersion at very high n_g values within the bandwidth of the slow-light transmission regime [10]. This paper focuses on experimental characterization of GVD, third-order dispersion (TOD), and fourth-order dispersion (FOD) in PhC waveguides by utilizing an integrated Mach-Zehnder interferometer (MZI) consisting of a photonic wire in one arm and a PhC waveguide in the second arm.

2. Experiment

The photonic-crystal structures were created by fabricating a periodic array of holes in a triangular lattice (lattice constant a=437nm) on 200mm silicon-on-insulator (SOI) wafers with 2µm thick buried oxide (BOX) and 220 nm thick silicon layers [11, 12]. The W1 PhC waveguides were formed by removing a row of holes in the Γ-K direction of the lattice. Polymer cladding was defined on the inverted taper spot-size converter for efficient fiber coupling [11, 12]. To optically characterize the devices, light from a broadband LED source (1200 to 1700nm) was coupled to the polarization maintaining (PM) fiber, transferred through the polarization controller, and finally launched into the device with a microlensed PM fiber tip. Transmission was measured with an optical spectrum analyzer (OSA) for the TE-like polarization (E-field parallel to the slab plane) with a resolution ranging from 5nm to 0.06nm.

The integrated MZI structure shown by the scanning electron microscope (SEM) images in Fig. 1 was fabricated to obtain accurate phase information in order to characterize higher order dispersion properties. One arm of the MZI consisted of a 250µm long W1 PhC waveguide, while the second arm has a photonic wire of the same length. 15o-angle Y-splitters were employed to equally distribute the input signal into the two arms and finally recombine them into the output waveguide [13]. The measured transmission spectra through the MZI were normalized on transmission through a reference straight strip waveguide, hence minimizing spurious Fabry-Perot oscillations and coupling noise. Measurements were conducted first with a coarse resolution (5nm-2nm) over a broad wavelength range. When dense interference fringes were observed, the spectrum was scanned again after increasing the OSA resolution to 0.06nm and narrowing the wavelength to the relevant range; this technique makes possible observation of the minima and maxima of the individual fringes throughout the spectrum [4], even for reduced transmission amplitude where group velocity is below 0.01c. Hence, the MZI approach allows the maximization of signal-to-noise ratio by eliminating external phase distortions. For a more comprehensive analysis of high-order dispersion in the slow-light regime, devices with r/a ratio, where r is the radius of the holes, of around 0.25, 0.30 and 0.35 were fabricated and characterized.

 

Fig. 1. SEM image of a) an integrated MZI illustrating the input/output Y-splitters and a 10µm long W1 PhC waveguide in one arm, b) the input Y-splitter of an MZI structure with 250µm long PhC waveguide.

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3. Transmission measurements

The transmission through the PhC waveguide with r/a ratio of 0.25 is shown by the red spectrum in Fig. 2, while the integrated MZI transmission is illustrated by the black and blue spectra exhibiting interference fringes. The measured spectra are characterized by three main regimes. First, the wavelength range from 1500nm to 1550nm where the dispersion is linear results in a low-loss and flat transmission through the W1 waveguide. For the same wavelength regime, the measured MZI interference fringes have similar modulation depth of ~20dB and similar spectral separation between the peaks, indicating a fixed group index. Second, the W1 waveguide spectrum exhibits a large decrease in amplitude for wavelengths below 1500nm due to out-of-plane leakage when the guided mode crosses into the light-cone; as a result, the MZI transmission shows a large decrease in fringe visibility. Finally, a sharp transmission cut-off is visible at 1565nm for the W1 waveguide spectrum, characterizing the on-set of the even guided-mode. The wavelength region from 1550 to the cut-off is known as the “slow-light” regime due to the high group-index (small group velocity). The fringe density of the MZI transmission increases progressively in this slow-light regime, indicating rapid change in the relative phase-shift between the two arms caused by the sharp increase of the group-index in PhC waveguide.

Close to the transmission cut-off, the fringes converge and cannot be identified individually at resolution of 2nm. The blue spectrum in Fig. 2 illustrates the MZI transmission measured with 0.06nm resolution for wavelengths spanning from 1550nm to 1600nm. To further resolve the fringes above 1560nm and close to the transmission cut-off, another scan was performed from 1560nm to 1565nm with resolution of 0.06nm as shown by the inset in Fig. 2. The fringe separation changes from 20nm at ~1530nm to 0.1nm at ~1560nm. Beyond 1564nm, the wavelength dependent losses make the fringe visibility very small and unreliable for further analysis.

 

Fig. 2. The red spectrum is the transmission through a 250µm long W1 PhC waveguide. The black spectrum indicates the MZI transmission measured at 2nm resolution, while the dense blue spectrum was measured at 0.06nm resolution. The inset shows the interference fringes measured from 1560nm to 1565nm at 0.06nm resolution in order to distinguish the fringe maxima and minima close to the even guide-mode on-set.

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4. Results and discussion

4.1 Group-index

The experimental results for n_g in PhC waveguides with r/a ratio of 0.25, 0.30, and 0.35 are illustrated by solid circles in Fig. 3. In order to determine n_g, the peaks of individual fringes obtained from the MZI transmission measurement were identified. The high signal-to-noise ratio provided by the integrated MZI setup allows an accurate identification of the peaks throughout the slow-light regime. Then, the wavelength dependent n_g was calculated as: $n_g\left(\lambda \right)=\frac{{\lambda }_{min}{\lambda }_{max}}{2L\left({\lambda }_{min}-{\lambda }_{max}\right)}+n_{wg}$, [4, 14].

The experimental results show the on-set of the guided-mode becomes shifted to lower wavelengths for higher r/a ratio. Furthermore, the maximum n_g measured becomes smaller as r/a increases: the highest values measured are 110, 70, and 50 for r/a=0.25, 0.30, and 0.35 respectively. The decrease in the maximum n_g value measured for higher r/a ratio is due to coupling loss. When compared with the transmission spectra in Fig. 2, the measured n_g for r/a=0.25 shows an expected trend: n_g remains constant at ~4 in the linear regime and increases rapidly as the fringes become closely spaced in the slow-light regime of the guided mode.

 

Fig. 3. The blue, red, and green filled circles show the experimental group index for r/a=0.25, 0.30, and 0.35 respectively. The solid lines show the theoretical group-index values calculated with 3D plane-wave method following the fitting procedure described in the text.

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4.2 Group velocity dispersion (GVD)

GVD parameters were deduced by computing $\frac{{\lambda }^2}{2\pi c^2}\frac{\partial n_g}{\partial \lambda }$, the first derivative of the groupindex. The solid circles in Fig. 4(a) show the GVD parameters calculated from the n_g data obtained experimentally. The noise in the experimental n_g data results in noticeable jumps in GVD as shown in the figure. Because negative values obtained when taking the derivative of n_g are purely due to noise and do not have any physical meaning, only the positive GVD values are presented. For n_g as high as ~70, the three r/a values show similar GVD trend with a tight distribution. The measured GVD parameter is ~10^-2ps²/mm at n_g ~4, and ~10ps²/mm at n_g ~25; however, a large increase to 10²ps²/mm is observed at n_g ~100 for r/a=0.25.

4.3 High-order dispersion (TOD and FOD)

The TOD parameters were deduced by calculating $\frac{{\lambda }^2}{2\pi c}\frac{\partial \left(GVD\right)}{\partial \lambda }$, the first derivative of the GVD. The experimental TOD parameters illustrated in Fig 4(b) by filled circles exhibit similar trend and value for the three r/a values. Again, negative values due to GVD noise are omitted in order to present the data trend clearly. While TOD is ~10^-5ps³/mm for n_g ~4 and ~10ps³/mm for n_g ~25, it becomes as high as ~10⁴ps³/mm for n_g ~100.

In addition, Figure 4(c) shows the FOD parameter obtained by calculating $\frac{{\lambda }^2}{2\pi c}\frac{\partial \left(TOD\right)}{\partial \lambda }$. The filled circles and solid lines represent the positive values for FOD for the experimental and theoretical n_g data respectively. The FOD parameters deduced from the experimental data exhibit similar trend for r/a=0.25, 0.30, and 0.35. FOD is ~10^-5ps⁴/mm for n_g ~4 and ~10ps⁴/mm for n_g ~25, but becomes as high as ~10⁵ps₄/mm for n_g ~100.

4.4 Theoretical fitting

To compare the experimental results with theory, 3D plane-wave calculations were performed for TE-like modes using the MIT photonic-band code [15]. The number of plane-waves was set to 16x16x16 and the tolerance for eignvalue convergence was set to 10^-8. Furthermore, 30 k-points were interpolated for wave-vectors between 0.25 and 0.50 in the Γ-X direction in order to focus on the regime of interest with high accuracy. After solving for 65 bands, n_g was deduced by performing the derivative $\left(\frac{1}{c}\frac{\partial w}{\partial k}\right)$ for the even guided-mode. GVD, TOD, and FOD were calculated by taking higher-order derivatives of n_g as explained in previous sections.

In order to get the best fitting to the experimental data, the following procedure was applied. First, the initial input parameters into the calculation, such as radius of holes (r), height of the slab (h), and the lattice constant (a), were determined from SEM measurements. Typically, these parameters are known with at most ±5% accuracy [14]. The calculations were iterated while varying the values for h, r, and a within 5% interval until the wavelength for the on-set of the even guided-mode matched well with the experimentally obtained transmission cutoff. Second, the input values were refined until a good fit was found between the slope of the calculated even guided-mode and the experimentally measured group-index. Finally, the input values were further refined until the best fit was found for GVD, TOD, and FOD while maintaining the good fit for wavelength and group-index. Figure 3 illustrates the best theoretical fits obtained through the above procedure.

A good agreement between the experimental (filled circles) and theoretical n_g data (solid lines) is shown in Fig. 3. The theoretical results were blue-shifted by ~3% to fit to the experimental data throughout the linear and slow-light regime. A slight offset between experimental and theoretical data can be observed as n_g increases rapidly when transitioning from the linear to the slow-light regime. The theoretical n_g values above ~50 are slightly redshifted when compared to the experimental data.

Furthermore, the solid lines in Fig. 4(a) illustrate the GVD parameters calculated from the theoretical data. PhCs with r/a=0.30 and 0.35 exhibit almost identical GVD, while r/a=0.25 shows lower GVD for n_g values above 25. There is a good fit between the experimental and theoretical GVD for all r/a values, especially until n_g ~25. Even though there is a matching trend for n_g >25, the theoretical GVD results are smaller than those deduced from the experimental data.

The TOD and FOD parameters calculated from the theoretical n_g are shown by the solid lines in Fig. 4(b) and Fig. 4(c) respectively. The figure indicates a good fit for TOD between theory and experiment until n_g ~15, beyond which there is an increasing deviation. However, the FOD parameters deduced for the calculated n_g data do not fit the experimental data even for low n_g values. For n_g ~100, the theoretical fit underestimates the experimental value by as high as three orders of magnitude.

The fitting procedure described above captures the best set of geometrical parameters of the structure. The large discrepancy is probably due to the complex nature of the mode at the edge of the Brillouin zone. Hence, most likely much larger number of plane-waves would be required in order to completely resolve the dispersion and get better fitting of higher-order dispersion. However, this requires excessively long calculation time. This finding also indicates the need to pay special attention when considering high-order dispersion parameters obtained only from theoretical calculations.

 

Fig. 4. a) Group-velocity dispersion (GVD). b) Third-order dispersion (TOD). c) Fourth-order dispersion (FOD). The blue, red, and green filled circles indicate experimental data for r/a=0.25, 0.30, and 0.35 respectively. The solid lines show best fit to experimental data according to fitting procedure described in the text.

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5. Conclusion

For the first time to our knowledge, high-order dispersion up to fourth order in W1 PhC waveguides has been measured for group-index values as high as 110. Highly accurate measurements in the slow-light regime were performed with integrated MZI structures which maximized signal-to-noise ratio by eliminating external phase distortions. For instance, the experimental value for GVD, TOD, and FOD parameters are ~10²ps²/mm, ~10⁴ps³/mm, and ~10⁵ps³/mm respectively for n_g ~100. The results provide benchmark propagation constants useful for evaluating pulse broadening due to high-order dispersion in the slow-light regime for W1 waveguides. The results are also important for evaluating non-linear effects that are dependent on high-order dispersion. The experimental results were compared with theoretical results obtained from 3D plane-wave calculations. While good agreement was found between theory and experimental results for n_g and GVD, large deviation was observed for TOD and FOD even at low n_g values. This indicates the need for further understanding of theoretical fitting of high-order dispersion. To summarize, the results illustrate the importance of considering higher-order dispersion parameters at high n_g values for W1 PhC waveguides.