## Abstract

We present a process calibration method for designing silicon-on-insulator (SOI) contra-directional grating couplers (contra-DCs). Our method involves determining the coupling coefficients of fabricated contra-DCs by using their full-width-at-half-maximum (FWHM) bandwidths. As compared to the null method that uses the bandwidth measured at the first nulls, our FWHM method obtains more consistent results since the FWHM bandwidth is more easily determined. We also extract the coupling coefficients using curve-fitting which provide values that are in general agreement with the values obtained using our method. However, as compared to the curve-fitting method, our method does not require knowledge of the insertion loss and is easier to implement. Our method can be used to predict the FWHM bandwidths, the maximum power coupling factors, the minimum power transmission factors, and the through port group delays and dispersions of subsequent, fabricated devices, which is useful in designing filters.

© 2015 Optical Society of America

## 1. Introduction

In communication applications that involve multiplexing and/or demultiplexing optical signals, maximizing the number of usable channels is essential for creating high data-rate interconnects [1–3]. Silicon contra-directional grating couplers (contra-DCs) are particularly useful in optical filtering applications because they do not have periodic spectral responses like ring resonator-based filters [4–10]. Silicon contra-DCs have been experimentally demonstrated in numerous publications [4–19]. Although previous demonstrations of silicon contra-DCs have shown good results, it remains challenging to design a filter’s bandwidth and have the “as-fabricated” device’s bandwidth, maximum power coupling factor, and minimum power transmission factor correspond to the design values, in the presence of lithography smoothing [20, 21]. Using a calibration procedure for the design process, a filter designer would be able to design a contra-DC such that the “as-designed” spectra closely matches the as-fabricated spectra. In this paper, we present a process calibration method which can be used to determine the absolute value of the coupling coefficient, |*κ*|, of a fabricated contra-DC by measuring its full-width-at-half-maximum (FWHM) bandwidth, Δ*λ _{bw}*. Once |

*κ*| is known, the through port and drop port spectra can be simulated. We demonstrate the effectiveness of our FWHM method (similar to [22, 23]) by extracting the |

*κ*|s of contra-DCs that were fabricated using electron beam lithography [24] on three fabrication runs. Our FWHM method for extracting |

*κ*| provides more consistent results as compared to using the null bandwidth (see [20, 21, 25–27]) due to the fact that Δ

*λ*can be more easily determined. Also, as compared to using the null method, the |

_{bw}*κ*|s extracted using our FWHM method are in general agreement with the values extracted by curve-fitting the drop port spectra. We then show that, using our FWHM method to extract |

*κ*|, the simulated spectra agree well with the experimental spectra. Also, the simulated through port group delay and dispersion responses of a particular device are calculated using the extracted |

*κ*|, which agree well with the Hilbert transform-determined and the measured group delay and dispersion responses.

## 2. Contra-DC theory and process calibration method

First, we will discuss the theoretical aspects of contra-DCs and the contra-DC design we used in this paper. The contra-DC design [Figs. 1(a) and 1(b)] has two strip waveguides, waveguide “a” and waveguide “b,” which have different average waveguide widths, w_{a} and w_{b}, respectively [9]. The waveguides have the same height and are separated from each other by an average gap distance, g [9]. Each waveguide has periodic grating corrugations, with a grating period, Λ, defined in Fig. 1(b), on the sidewalls located within the gap region [9]. The corrugation widths are labelled c_{a} and c_{b} for waveguide “a” and waveguide “b,” respectively [9]. The corrugations allow the coupler to act as a Bragg reflector with the strength of the inter-waveguide coupling determined by the inter-waveguide coupling coefficient, *κ* [7,9,16]. We have also included anti-reflection gratings on the external sidewalls of the waveguides to suppress the intra-waveguide Bragg reflections [7, 9, 16, 28].

The power transferred from the input port to the drop port of a contra-DC is given by the power coupling factor, |*κ _{c}*|

^{2}, and the amount of power transferred to the through port is given by the power transmission factor, |

*t*|

_{c}^{2}, which can be calculated using the following equations,

*β*and

_{a}*β*are the propagation constants of waveguide “a” and waveguide “b” without corrugations with widths equal to w

_{b}_{a}and w

_{b}, respectively,

*L*is the length of the coupler, and

*m*is the grating order which equals 1 since we are using first-order gratings [29]. Equation (1) is the same as Eq. (13.5–19) in [29] and Eq. (1) in [4] and Eq. (2) can be determined from Eq. (13.5–16) in [29]. In this paper,

*β*and

_{a}*β*are calculated by numerically determining the wavelength dependent effective indices of the waveguides using MODE Solutions by Lumerical Solutions, Inc., and curve-fitting them to third-order polynomials [9]. The material model that was used for silicon included dispersion and was loss-less [2, 9, 21, 30] and the refractive index for silicon dioxide was fixed at 1.4435 [2, 9, 30]. The inter-waveguide coupling coefficient,

_{b}*κ*, is defined as the strength of the coupling of light from waveguide “a” to waveguide “b” within the contra-DC and can be calculated using the following equation [4, 11, 13, 15, 16, 28, 29],

*ω*is the angular frequency,

*ξ*(

_{a}*x*,

*y*) is the transverse mode of waveguide “a,”

*ξ*(

_{b}*x*,

*y*) is the transverse mode of waveguide “b,” and

*ε*(

_{m}*x*,

*y*) is the

*m*

^{th}component of the Fourier series expansion of the dielectric perturbation. Using this equation, two different methods have been used to calculate |

*κ*| [16, 28]. The first method involves treating each waveguide as isolated [4, 16, 28, 29, 31, 32]. In this method,

*ξ*(

_{a}*x*,

*y*) and

*ξ*(

_{b}*x*,

*y*), correspond to the modes of the isolated unperturbed waveguides [4,16,28,29,31,32]. The second method involves calculating the first and second-order transverse modes of the coupler (

*i.e.*, supermode theory) [10, 13, 15, 16, 28, 32]. [20] and [21] have demonstrated that there is a large difference between the modeled results and experimental results for SOI Bragg gratings (see Fig. 2.35 in [20] and Fig. 4.43 in [21]). [4] showed good agreement between experimental results and simulated results for contra-DCs by using Eq. (3). However, cross-sectional SEM images were needed for calibration. Recently, [27] demonstrated a method to model Bragg gratings using 3-D finite-difference time-domain (FDTD) simulations and Bloch boundary conditions which showed good agreement between theoretical and experimental results. However, the above-mentioned methods require knowledge of the effects of the lithography on the shape of the grating. One method to significantly reduce the difference between the modeled results and the experimental results is to take into account how the fabrication process affects the design of the device (

*e.g.*, lithography smoothing) by using lithography simulation software, such as Mentor Graphics Calibre, and then simulating the structure using 3-D FDTD simulation software, such as FDTD Solutions by Lumerical Solutions, Inc., [20, 21]. However, this process is more complex since knowledge of fabrication process parameters are needed. In this paper, we will demonstrate an experimental method to determine |

*κ*| by using Δ

*λ*. With our experimental method, we can extract |

_{bw}*κ*| without having to measure the effects of lithography directly.

There are three steps to extract |*κ*|. The first step is to determine Δ*λ _{bw}*, which can be measured directly from the drop port spectrum. The second step is to determine the average propagation constant mismatch,

*δβ*. To obtain

_{avg}*δβ*, we use the propagation constant differences,

_{avg}*δβ*and

_{H}*δβ*, where

_{L}*δβ*is measured from the frequency that corresponds to the center of the main lobe to the high-frequency half-maximum point and

_{H}*δβ*is measured from the frequency that corresponds to the center of the main lobe to the low-frequency half-maximum point (for a complete mathematical description see Appendix A). The magnitude of

_{L}*δβ*and the magnitude of

_{H}*δβ*are given in Eqs. (4) and (5), respectively, and defined graphically in Fig. 2(a),

_{L}*f*=

_{H}*f*−

_{H}*f*

_{0}, Δ

*f*=

_{L}*f*

_{0}−

*f*, Δ

_{L}*λ*=

_{L}*λ*

_{0}−

*λ*, Δ

_{L}*λ*=

_{H}*λ*−

_{H}*λ*

_{0},

*f*

_{0}and

*λ*

_{0}are the frequency and wavelength corresponding to the center frequency and center wavelength (middle point between the FWHM points), respectively,

*f*and

_{H}*λ*correspond to the higher frequency FWHM point and the lower wavelength FWHM point, respectively,

_{L}*f*and

_{L}*λ*correspond to the lower frequency FWHM point and the higher wavelength FWHM point, respectively,

_{H}*n*and

_{g,a}*n*are the group indices of waveguide “a” and waveguide “b,” respectively, and

_{g,b}*c*is the speed of light in a vacuum. Equations (4) and (5) are similar to Eq. (13.5–22) in [29] but here we include the effects of dispersion and use the group indices, since dispersion affects the spectral response of contra-DCs (see [33]). To determine

*δβ*, we take the average of Eqs. (4) and (5),

_{avg}*λ*=

_{bw}*λ*−

_{H}*λ*. Here, for convenience, the wavelength dependent group indices are numerically determined using MODE Solutions by Lumerical Solutions, Inc., and curve-fitted to third-order polynomials. Equation (6) is similar to Eq. (31) in [22] but here we include the effects of dispersion and use the group indices. Figure 2(b) shows the experimental drop port spectrum of one of our fabricated contra-DCs, with a gap distance of 140 nm, as a function of Δ

_{L}*β*. The FWHM becomes 2

*δβ*when the spectrum is plotted as a function of Δ

_{avg}*β*. By plotting our spectral response as a function of Δ

*β*, we are able to directly measure

*δβ*from the spectral response without having to use Eq. (6). One may use either of these methods to determine

_{avg}*δβ*but we will focus on the method that utilizes Eq. (6).

_{avg}The third step is to extract |*κ*| by using Eq. (1) and Eq. (6). We replace Δ*β* in Eq. (1) with Eq. (6) as shown in the left-hand side of Eq. (7) where *s*^{2} = |*κ*|^{2} − (*δβ _{avg}*/2)

^{2}. Since Eq. (1) reduces to tanh

^{2}(|

*κ*|

*L*) for Δ

*β*= 0 [29], we can find the FWHM intensity by dividing tanh

^{2}(|

*κ*|

*L*) by 2 and finding the value of |

*κ*| that will satisfy Eq. (7) for our value of

*δβ*. [22, 23] use a similar method to extract the bandwidths of contra-DCs.

_{avg}*κ*| that is largest.

An alternative method, using the nulls to determine |*κ*|, is to measure the bandwidth at the first nulls to the left and to the right of the main lobe and to use Eq. (8) (similar to [20,21,25,34] and is a re-arrangement of the equation found in [26, 29]),

*δβ*is calculated using Eq. (6) but using the first null points instead of the FWHM points. [21, 27] have also used the null bandwidth to extract |

_{avg}*κ*| but for SOI Bragg gratings. Also, another method to extract |

*κ*| is to curve-fit the drop port spectrum of the contra-DC using a nonlinear least-squares method. As we will show in the next section, the extracted |

*κ*|s from the curve-fit method and from the FWHM method are in general agreement with each other as compared to the values determined using the null method. However, the curve-fit method relies on an accurate normalization of the measured drop port spectrum (an issue which others have previously mentioned [35]) whereas the FWHM method does not require that the measured data be normalized. Both the FWHM method and the curve-fit method provide more consistent results than the null method does. Also, provided that

*δβ*can be accurately obtained, our FWHM method should be applicable to devices fabricated in other material platforms because the method is not platform dependent.

_{avg}## 3. Experimental results

Electron beam lithography was used to fabricate the SOI contra-DCs [24] and a silicon dioxide cladding layer was deposited on top of the devices. The silicon strip waveguide heights were all chosen to be 220 nm. The width of waveguide “a” was 450 nm and the width of waveguide “b” was 550 nm. The corrugation widths for waveguide “a” and “b” were 30 nm and 40 nm, respectively. These dimensions were taken from [9]. The grating period was chosen to be 312 nm and the number of periods was chosen to be 500. Therefore, the total length of each contra-DC was 156 μm. The gap distances were varied between 120 nm and 400 nm in 20 nm increments for a total of 15 devices. Fiber grating couplers were used for coupling light into and out of the devices [36, 37]. The contra-DCs were fabricated on three separate fabrication runs, “run 1,” “run 2,” and “run 3” at different times. Fully-etched fiber grating couplers [37] were used in “run 1” and “run 3” and shallow-etched fiber grating couplers [36] were used in “run 2.” The experimental drop port spectra of four of the devices from “run 1,” “run 2,” and “run 3” with gap distances equal to 140 nm, 220 nm, 340 nm, and 400 nm are shown in Figs. 3(a), 3(c), and 3(e), respectively. The experimental through port spectra of four of the devices from “run 1,” “run 2,” and “run 3” with gap distances equal to 140 nm, 220 nm, 340 nm, and 400 nm are shown in Figs. 3(b), 3(d), and 3(f), respectively. The fiber grating coupler response was removed from both the through port and drop port spectral responses by normalizing the spectra to the fiber grating response envelope in the through port spectral response.

The relationship between the bandwidths of contra-DCs and their gap distances has been theoretically [4,11] and experimentally [4,5] demonstrated, and shows that, as the gap distance increases, the bandwidth decreases. Also, the relationship between |*κ*| and the gap distance has been theoretically demonstrated, and shows that, as the gap distance increases, |*κ*| exponentially decreases [10]. Here, we also experimentally demonstrate the relationship between Δ*λ _{bw}* and the gap distance, which is in agreement with previously published results. Also, we experimentally demonstrate the relationship between |

*κ*| and the gap distance, which is in agreement with the theoretical results in [10]. Figures 4(a) and 4(b) show Δ

*λ*and the extracted |

_{bw}*κ*| (extracted using our FWHM method) versus gap distance, respectively, for the contra-DCs fabricated on “run 1,” “run 2,” and “run 3.” As the gap distance increases, Δ

*λ*and |

_{bw}*κ*| tend to decrease and Δ

*λ*reaches a minimum and for one of our devices, the device from “run 3” with a gap distance of 400 nm, we are not able to obtain a value for |

_{bw}*κ*| since it goes to zero. We also fabricated contra-DCs on “run 1” with a fixed gap distance of 280 nm and varied the corrugation widths of waveguide “a” and waveguide “b.” Figures 4(c) and 4(d) show Δ

*λ*and the extracted |

_{bw}*κ*| versus corrugation width, respectively, for corrugation widths of 30 nm to 150 nm in 20 nm increments for waveguide “a” and corrugation widths of 40 nm to 160 nm in 20 nm increments for waveguide “b.” As the corrugation width increases, Δ

*λ*[4, 6, 11] and |

_{bw}*κ*| [10] increase.

Next, we provide a comparison between the |*κ*|s extracted using the FWHM method [using Eqs. (6) and (7)], the null method [using Eqs. (6) and (8)], and the curve-fit method (using MATLAB’s *lsqcurvefit* function [38]) from the devices made in three fabrication runs. Figures 5(a), 5(b), and 5(c) show the extracted |*κ*|s using the three methods for “run 1,” “run 2,” and “run 3,” respectively. Upon inspection of Figs. 5(a)–5(c), it is clear that the |*κ*|s that were determined using the FWHM method and the curve-fit method exhibit nearly exponential trends, as expected. The |*κ*|s extracted using the FWHM method and the curve-fit method are relatively close to each other as compared to the |*κ*|s determined using the null method. The discrepancies seen in Figs. 5(a)–5(c) using the null method are due to the difficulty in determining the locations of the nulls [*e.g.*, see Fig. 5(d)]. Also, we were unable to determine the |*κ*|s for five of the devices using the null method since there are no valid solutions to Eq. (8). For one of the devices using the FWHM method we could only extract a zero solution for |*κ*|. With the curve-fit method, we were able to extract a non-zero value for |*κ*| for each of the devices.

Next, we demonstrate, for a given contra-DC with a fixed coupling length, that Δ*λ _{bw}* reaches a minimum value as |

*κ*| approaches zero ([34] also demonstrated this trend in Bragg gratings). To determine the theoretical minimum bandwidth, Δ

*λ*, the following equation can be used (see Appendix B for the derivation),

_{bw−min}*λ*changes as the coupling length increases (the group indices were evaluated at 1535.33 nm). Δ

_{bw−min}*λ*can be reduced by increasing the coupling length [39]. The device from “run 3” with a gap distance of 400 nm has a measured bandwidth below Δ

_{bw−min}*λ*(due to the experimental results having ripples likely caused by the grating couplers), which could be the reason that there is no |

_{bw−min}*κ*| solution other than zero for this device using the FWHM method.

Next, we show an example of using the extracted |*κ*| (using the FWHM method) to closely match the simulated spectra to the experimental spectra of one of our contra-DCs. We have chosen one of our devices that showed a highly symmetric spectral response to the left and right of the center of the main lobe for the comparison between the simulated results (using the extracted |*κ*| determined from the FWHM method) and the experimental results. The device has a gap distance of 140 nm and is from “run 2.” The simulated spectra were plotted using Eqs. (1) and (2) and we have added 0.0147 to the modeled values of the effective indices for spectral alignment purposes. Figure 7(a) shows that the simulated through port and drop port spectra using the extracted |*κ*| of 19882 m^{−1} from “run 2” closely match the experimental spectra. Figure 7(b) shows a comparison between the simulated spectra using the extracted |*κ*| of 18466 m^{−1} from “run 1” and the experimental spectra from “run 2.” The results in Fig. 7(b) show that, since there is close agreement between the two fabrication runs, using a previously extracted |*κ*| can be used to predict the spectral response of future fabricated devices with the same as-designed dimensions. Figures 7(c) and 7(d) show a comparison between the drop port spectra and through port spectra, respectively, from “run 1,” “run 2,” and “run 3” and the simulated spectra (we have aligned the measured spectra from “run 1,” “run 2,” “run 3,” and the simulated spectra to their respective center wavelengths) using the average |*κ*| of 18856 m^{−1}, calculated using the extracted |*κ*|s from the three runs (*i.e.*, 18466 m^{−1}, 19882 m^{−1}, and 18219 m^{−1}).

Our method can also be used to predict the maximum power coupling factors, MAX|*κ _{c}*|

^{2}s, and the minimum power transmission factors, MIN|

*t*|

_{c}^{2}s, of contra-DCs. In Figs. 8(a) and 8(b) we show a comparison between the experimental and simulated (using the extracted |

*κ*|s determined from the FWHM method) MAX|

*κ*|

_{c}^{2}s and MIN|

*t*|

_{c}^{2}s versus gap distance, respectively, for the devices from “run 1,” “run 2,” and “run 3.” Experimental MIN|

*t*|

_{c}^{2}s for the devices from “run 1” with gap distances of 320 nm, 380 nm, and 400 nm and for the devices from “run 3” with gap distances of 380 nm and 400 nm are not shown in Fig. 8(b) since the main notches within their through port spectra were not visible. Simulated MAX|

*κ*|

_{c}^{2}and MIN|

*t*|

_{c}^{2}for the device from “run 3” with a gap distance of 400 nm is not shown since we were unable to extract a value for |

*κ*| other than zero. Also, the MAX|

*κ*|

_{c}^{2}s determined using the curve-fit method are closer to the normalized measured results as compared to the MAX|

*κ*|

_{c}^{2}s determined using the FWHM method. However, using the |

*κ*|s extracted by the FWHM method result in many of the simulated MIN|

*t*|

_{c}^{2}s being closer to the measured results as compared to the MIN|

*t*|

_{c}^{2}s determined using the curve-fit method. The likely reason that the FWHM method gives better results for MIN|

*t*|

_{c}^{2}, as compared to the values determined using the curve-fit method, is that the curve-fit method relies on an accurate normalization of the drop port spectrum.

The group delay and the dispersion of a contra-DC is of interest because they give us an indication of the effect the contra-DC will have on a signal. Previously, it has been shown that the phase (and, therefore, the group delay and dispersion) of fiber Bragg gratings [40–42] and ring resonators [43–46] can be determined using the Hilbert transform method. Specifically, the Hilbert transform method can be used to determine through port phase responses of Bragg gratings because the through port response is minimum phase [40–42]. Here, we use the Hilbert transform method [41] to determine the through port phase of a contra-DC from “run 2” with a gap distance of 140 nm (we use the *hilbert* function from MATLAB [47]). Once the phase response is determined, the group delay [9, 48] and dispersion [9, 48, 49] can be calculated. Figures 9(a) and 9(b) show the group delay and dispersion responses, respectively, using the Hilbert transform method on the experimental through port spectrum from Fig. 7(a) (the results shown were smoothed using moving averages) and are compared to the simulated responses that were determined for the |*κ*| extracted using the FWHM method and the measured results (the average of 300 measurements) using an Optical Vector Analyzer™ ST*e* by Luna Innovations, Inc., (OVA). For the simulated results, we added an additional phase to account for the transit time of the device. The effective indices for this additional phase were calculated for a waveguide width of 450 nm using MODE Solutions by Lumerical Solutions, Inc. Similarly, we have added a 2.22 ps group delay offset to the Hilbert transform-determined group delay. Also, a constant group delay offset was subtracted from the measured group delay for alignment to the simulated result. The Hilbert transform-determined through port group delay and dispersion results are in close agreement with the simulated results using the extracted |*κ*| and the measured results using the OVA. Therefore, our FWHM method can also be used to predict the through port group delay response and the dispersion response of contra-DCs.

## 4. Conclusion

In the filter design process, the ability to predict the performance of contra-DCs is invaluable. We have presented a method, the FWHM method, for determining the coupling coefficients of contra-DCs. To demonstrate the usefulness of our method, we fabricated SOI contra-DCs on three separate fabrication runs. Our FWHM method of extracting the coupling coefficient of contra-DCs can be used to predict the spectral response, group delay, and dispersion of subsequently fabricated devices. The FWHM method provides more consistent extracted coupling coefficient values as compared to the values extracted using the null method. Also, the FWHM method provides extracted coupling coefficient values of fabricated devices that are relatively close, as compared to using the null method, to the values extracted by curve-fitting the drop port spectra. However, the curve-fit method relies on the accurate normalization of the drop port spectrum whereas our FWHM method does not require normalization of the data, and our method is generally easier to implement. We have also shown that there is a minimum bandwidth that can be obtained by reducing the coupling coefficient, which needs to be considered when designing a contra-DC-based filter. We have presented an equation for this minimum bandwidth as a function of the length of the coupler. The method presented in this paper can be used to calibrate the design process, enabling designers to accurately predict the as-fabricated filter response.

## Appendices

## A. Derivation of the average propagation constant mismatch

Here, we will derive the equation for the average propagation constant mismatch, *δβ _{avg}*. The first step in determining

*δβ*is to calculate the propagation constant difference,

_{avg}*δβ*, which is defined as the difference between the propagation constant mismatch, Δ

_{H}*β*(

*f*), at the frequency,

_{H}*f*, corresponding to the intensity at FWHM at the higher frequency and the propagation constant mismatch, Δ

_{H}*β*(

*f*

_{0}), at the center frequency,

*f*

_{0}, (see Fig. 10) as shown in Eq. (10).

*β*and

_{a}*β*are the propagation constants of waveguide “a” and waveguide “b” in isolation, respectively,

_{b}*m*is the grating order which is equal to 1 for our first-order contra-DCs, and Λ is the grating period [29] as shown in Eq. (11).

*n*and

_{a}*n*are the effective indices of waveguide “a” and waveguide “b”, respectively, and

_{b}*c*is the speed of light in a vacuum. Since the effective indices are frequency dependent due to dispersion, we will express

*n*(

_{a}*f*) as ${n}_{a}({f}_{0})+\mathrm{\Delta}\hspace{0.17em}{{f}_{H}\frac{\text{d}{n}_{a}}{\text{d}f}|}_{{f}_{0}}$ and

_{H}*n*(

_{b}*f*) as ${n}_{b}({f}_{0})+{\mathrm{\Delta}{f}_{H}\hspace{0.17em}\frac{\text{d}{n}_{b}}{\text{d}f}|}_{{f}_{0}}$ where Δ

_{H}*f*=

_{H}*f*−

_{H}*f*

_{0}[50]. After simplification by grouping terms,

*δβ*becomes,

_{H}*f*<<

_{H}*f*

_{0}[50]. Therefore, the final equation for

*δβ*is,

_{H}*n*and

_{g,a}*n*are the group indices of waveguide “a” and waveguide “b”, respectively.

_{g,b}Next, we show the equation for the propagation constant difference, *δβ _{L}*, [see Eq. (17) where Δ

*f*=

_{L}*f*

_{0}−

*f*] which is defined as the difference between the propagation constant mismatch, Δ

_{L}*β*(

*f*), at the frequency,

_{L}*f*, corresponding to the intensity at FWHM at the lower frequency and the propagation constant mismatch, Δ

_{L}*β*(

*f*

_{0}), at the center frequency,

*f*

_{0}(see Fig. 10).

*δβ*was derived using the same procedure as used to derive

_{L}*δβ*.

_{H}*δβ*,

_{avg}*λ*

_{0},

*λ*, and

_{L}*λ*correspond to the frequencies

_{H}*f*

_{0},

*f*, and

_{H}*f*, respectively, and Δ

_{L}*λ*=

_{bw}*λ*−

_{H}*λ*. Equations (16) and (17) are similar to Eq. (13.5–22) in [29] and Eq. (22) is similar to Eq. (31) in [22] except that we have taken dispersion into account.

_{L}## B. Derivation of the minimum bandwidth

Here, we present the derivation for the minimum bandwidth of a contra-DC, Δ*λ _{bw−min}* as |

*κ*| goes to zero. First, we rearrange the terms in Eq. (7) as shown below,

*δβ*= 2.783115 (we neglect the trivial solution which is zero). Therefore, for a contra-DC with a given

_{avg}L*L*,

*δβ*needs to be greater than 2.783115. Therefore, substituting

_{avg}L*δβ*= 2.783115 into Eq. (22), we get (similar to [39]), which is approximately equal to,

_{avg}L## Acknowledgments

We would like to acknowledge the Natural Sciences and Engineering Research Council (NSERC) of Canada and the SiEPIC program for their support. We also acknowledge CMC Microsystems, Lumerical Solutions, Inc., and Mentor Graphics for the software products that were used in this project. We would like to thank Richard Bojko for fabrication of the devices, Dr. Wei Shi for technical help and insightful discussions, Miguel Ángel Guillén Torres for insightful discussions, Yun Wang for the design of the fiber grating couplers, Han Yun for layout support, Jonas Flueckiger for technical help with Pyxis by Mentor Graphics, and Fan Zhang for measurements of some of the fabricated devices. Part of this work was conducted at the University of Washington Nanofabrication Facility, a member of the NSF National Nanotechnology Infrastructure Network.

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