## Abstract

We design silicon ridge/rib waveguide directional couplers which are simultaneously tolerant to width, height, coupling gap, and etch depth variations. Using wafer-scale measurements of structures fabricated in the IMEC Standard Passives process, we demonstrate the normalized standard deviation in the per-length coupling coefficient (a metric for the splitting ratio variation) of the variation-tolerant directional couplers is up to 4 times smaller than that of strip waveguide designs. The variation-tolerant couplers are also the most broadband and the deviation in the coupling coefficient shows the lowest spectral dependence.

© 2014 Optical Society of America

## 1. Introduction

Silicon-on-insulator (SOI) is attracting significant interest for integrated optics because of its compatibility with standard CMOS fabrication processes [1–3]. The large index contrast between the core and cladding in SOI waveguides strongly confines light to submicron dimensions, which can enable densely integrated photonic circuits. However, a significant drawback to the high index contrast is that the optical properties of devices become sensitive to nanometer-scale dimensional variations [4, 5]. In a 248 nm or 193 nm deep UV photolithography fabrication process, dimensional variations can be tens of nanometers [6, 7], which can reduce device yields or increase tuning powers [3,5,8,9]. At the photonic circuit level, tuning power and variation tolerance can be improved using multiple devices or compound devices [5, 10]; however, the basic elements from which more complex devices are formed can also be optimized.

A common element that is sensitive to dimensional variation is the directional coupler. For a “standard” 3 dB directional coupler formed with 500 nm wide, 220 nm tall SOI strip waveguides separated by a 200 nm gap, we calculate that the splitting ratio changes by 0.7% per 1 nm change in the gap. This sensitivity is especially detrimental in interferometer and microring modulators and filters, where achieving specific splitting ratios is critical to optimal performance [11–15].

In this work, we show that SOI ridge (sometimes referred to as “rib”) directional couplers can be designed to have splitting ratios that are simultaneously insensitive to variations in waveguide width, height, coupling gap, and etch depth. Especially important is the tolerance to height and etch depth, which affect the modal effective index more strongly than the waveguide width [4, 7]. We characterize directional coupler designs using wafer-scale measurements of Mach-Zehnder interferometer test structures implemented in a IMEC Standard Passives multi-project-wafer shuttle, which uses 193 nm DUV lithography and 200 mm SOI wafers. The sensitivity of the per-length coupling coefficient of the variation-tolerant design is up to 4 times smaller than that of standard strip coupler designs. The coupling coefficient and the normalized standard deviation of the variation-tolerant design also exhibited the least spectral dependence.

## 2. Sensitivity analysis and design of coupled waveguides

To motivate the design, we first analyze the dimensional sensitivity of a pair of coupled waveguides. Figure 1 shows a pair of symmetric, coupled waveguides and the relevant geometric parameters. The width of the waveguide core is *w*; the coupling gap is *g*; the waveguide height is *h*; and the slab thickness is *t*. The fraction of cross-coupled power, *K*, is

*n*is the effective index difference between the symmetric and antisymmetric super-modes and

*L*is the length of the coupler. If

*α*is an arbitrary geometric parameter (i.e.,

*α*can be replaced by

*w*,

*g*,

*h*, or

*t*), for the shortest coupler to achieve

*K*,

*K*is characterized by a sensitivity parameter, $\frac{\partial \mathrm{\Delta}n}{\partial \alpha}/\mathrm{\Delta}n$, so a robust directional coupler would have Δ

*n*stationary with respect to as many dimensional variations as possible.

In a symmetric directional coupler, Δ*n* is stationary with respect to asymmetric variations in waveguide widths, so
$\frac{\partial K}{\partial \alpha}$ is only affected by a common change in *w* in both waveguides. An increase in *w* would increase the mode confinement, which reduces Δ*n*. However, Δ*n* is also affected by the gap between the waveguides, and a decrease in *g* increases the mode overlap between the two waveguides and Δ*n*. Therefore, if an increase in *w* is perfectly correlated with an equal and opposite decrease in *g* (i.e., statistical correlation of *ρ _{w,g}* = −1), it is possible to balance an increase in waveguide proximity by an increase in mode confinement to achieve

*g*and

*w*implies the width variations are symmetric with respect to the waveguide centerlines. This is an approximation to actual lithography and/or etching imperfections, which can also cause asymmetric variations.

Figure 2(a) shows the simulated sensitivity parameter
$\left(\frac{\partial \mathrm{\Delta}n}{\partial w}-\frac{\partial \mathrm{\Delta}n}{\partial g}\right)/\mathrm{\Delta}n$ for various values of *g* and *w* at a wavelength of 1550 nm and assuming nominal values of *h* = 220 nm, *t* = 150 nm, as fixed by the IMEC fabrication process. The calculations are for transverse electric (TE) modes and were done using Lumerical FDTD’s built-in eigenmode solver. Because of the lower index contrast,
$\frac{\partial \mathrm{\Delta}n}{\partial w}/\mathrm{\Delta}n$ and
$\frac{\partial \mathrm{\Delta}n}{\partial g}/\mathrm{\Delta}n$ of the ridge waveguide couplers are about 3 times smaller than that of strip waveguide couplers. For *g* ≳ 300 nm, *g* and *w* can be chosen such that Eq. (3) is satisfied. Additionally, at the design point *w* = *g* = 400 nm,
$\frac{\partial \mathrm{\Delta}n}{\partial h}$ and
$\frac{\partial \mathrm{\Delta}n}{\partial t}$ are also nearly zero, giving the most dimensionally tolerant design. Figures 2(b) and 2(c) show
$\frac{\partial \mathrm{\Delta}n}{\partial h}$ and
$\frac{\partial \mathrm{\Delta}n}{\partial t}$ for a few values of *w* and *g*.

There is no design for strip waveguide directional couplers (i.e., *t* = 0) which is similarly tolerant to all possible dimensional variations. For example, a standard strip waveguide directional coupler with *w* = 500 nm and *g* = 200 nm is nearly stationary with respect to correlated *w* and *g* changes, but
$\frac{\partial \mathrm{\Delta}n}{\partial h}/\mathrm{\Delta}n$ is −0.007 nm^{−1}.

## 3. Parameter extraction for measured devices

To test the variation tolerance of the directional coupler design, we implemented ridge and strip waveguide couplers in a IMEC Standard Passives run. Figure 3(a) shows the test structure, which consists of an unbalanced Mach-Zehnder interferometer (MZI), where the input 3 dB splitter is a multi-mode interference coupler (MMI) and the directional coupler to be tested is at the output end. Although MMIs tend to have higher insertion losses than directional couplers, they can more reliably achieve 50:50 splitting ratios by symmetric placement of the two output waveguides [16]. Assuming an ideal 50:50 split, *K* can be extracted from the extinction ratio (*ER*) of the spectral fringes of the MZI using [17]:

*K*and

*ER*at the output port as labelled in Fig. 3(a) are related by

*S*is the ratio between the output powers of the two MMI outputs.

To measure the devices, light from a tunable laser was fiber-coupled into and out of the chips using grating couplers. We measured directional couplers of different lengths and fit the data to *K* = sin^{2}(*κ _{L}L* +

*κ*

_{0}) to determine

*κ*, the per-length coupling coefficient, and

_{L}*κ*

_{0}, the residual coupling in the transition regions of the coupler. Figure 3(b) shows the transmission spectrum of a typical MZI test structure and the fringes. Because the parameter extraction only relies on

*ER*, the grating coupler bandwidth does not affect the extracted values of

*κ*. As shown in Fig. 4, the

_{L}*R*

^{2}values of the fits are better than 0.995. The agreement between the measured

*κ*and the theoretical value $\frac{\pi \mathrm{\Delta}n}{\lambda}$ from simulation was the best for the variation-tolerant ridge design. For ridge couplers, the agreement was within 0.2% for

_{L}*g*= 400 nm and 1% for

*g*= 500 nm. For the more sensitive strip couplers, the agreement was within 7% for

*g*= 250 nm, 15% for

*g*= 200 nm, and 25% for

*g*= 150 nm. The discrepancy is attributed to deviations between the nominal and fabricated dimensions, which become more pronounced as

*g*approaches the minimum allowed by the IMEC process.

## 4. Wafer-scale measurements

To characterize the wafer-scale variation tolerance, we measured test structures on 16 dies across the 200 mm SOI wafer. Figure 5 shows the wafer maps of *κ _{L}* at a wavelength of 1550 nm. From Eq. (2), an experimental measure of the coupler sensitivity is
$\frac{{\sigma}_{{\kappa}_{L}}}{{\kappa}_{L}}$, where

*σ*

_{κL}is the standard deviation of

*κ*. Table 1 summarizes the results. The quoted uncertainty includes uncertainty in the fitting and MMI splitting ratio. Using a separate test structure, we estimated a ±5% variation in the MMI splitting ratio, so

_{L}*S*in Eq. (5) varied from $\frac{45}{55}$ to $\frac{55}{45}$. We repeated the parameter extraction with these values of

*S*to obtain the uncertainties in the average

*κ*and

_{L}*σ*

_{κL}. The MMI splitting ratio causes the dominant source of uncertainty in

*κ*. The uncetainty in the

_{L}*ER*only contributes to an uncertainty of ±1% in the extracted values.

In terms of
$\frac{{\sigma}_{{\kappa}_{L}}}{{\kappa}_{L}}$, ridge couplers with *w* = *g* = 400 nm are more tolerant than the other strip and ridge couplers by a factor of about 2 at 1550 nm. The improvement may be limited by the imperfect anti-correlation between *g* and *w*. From the metrology data from 7 dies provided by IMEC, we calculated a statistical correlation coefficient of *ρ _{w,g}* = −0.55 between

*w*and

*g*, rather than −1 as assumed in our analysis.

Lastly, for the spectral characteristics of the coupler, we first note that the wavelength dependent change in the coupling ratio is given by

*K*from a representative die for several designs with similar values of

*K*. The variation-tolerant design has the lowest $\frac{\partial K}{\partial \lambda}$. The die-averaged extracted value of $\frac{\partial {\kappa}_{L}}{\partial \lambda}/{\kappa}_{L}$ is 1.3 ×10

^{−3}nm

^{−1}for this design, while it is 4.2 × 10

^{−3}nm

^{−1}for the

*g*= 200 nm strip waveguide design, agreeing well with simulations. Figure 6(b) shows the spectrum of $\frac{{\sigma}_{{\kappa}_{L}}}{{\kappa}_{L}}$. The spectral dependence of the normalized standard deviation is the lowest for the variation-tolerant design across the wavelength range from 1535 to 1565 nm, as limited by the grating coupler bandwidth. The maximum measured improvement in $\frac{{\sigma}_{{\kappa}_{L}}}{{\kappa}_{L}}$ of the variation-tolerant design compared to strip waveguide couplers is by about a factor of 4.

## 5. Conclusions

By exploiting correlations between the waveguide width and gap variations, we designed and tested silicon ridge waveguide directional couplers that are up to about 4 times more tolerant to wafer-scale variation than fully-etched couplers. Integrating variation-tolerant couplers within microring and interferometer devices can lead to more reliable performance, for example, in terms of the extinction ratio, passband flatness, and rejection ratio.

## Acknowledgments

Access to the IMEC Standard Passives MPW was supported by CMC Microsystems. We thank Dan Deptuck and Jessica Zhang of CMC Microsystems for their assistance. The support of the Natural Science and Engineering Research Council of Canada is gratefully acknowledged.

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