## Abstract

Over the last two decades, many filters requiring custom spectral responses were obtained from photo-inscribed fiber Bragg gratings because of the flexibility inherent to this technology. However, Bragg gratings in silicon waveguides have the potential to provide faster and more efficient tuning capabilities when compared to optical fiber devices. One drawback is that Bragg gratings filters with elaborate spectral amplitude and phase responses often require a long interaction length, which is not compatible with current integration trends in CMOS compatible photonic circuits. In this paper, we propose to make Bragg gratings in spiral-shaped waveguides in order to increase their lengths while making them more compact. The approach preserves the flexibility of regular straight grating structures. More specifically, we demonstrate 2-mm long gratings wrapped in an area of 200 µm x 190 µm without any spectral degradation due to waveguide curvature. Furthermore, we interleave three spiral waveguides with integrated gratings thereby tripling the density and demonstrate good phase compensation for each of them. Finally, we show that this approach is compatible with phase-apodization of the grating coupling coefficient.

©2013 Optical Society of America

## 1. Introduction

Optical filters with flexible and precisely tuned spectral responses are of great interest for many applications in communication and sensing. In the last two decades, fiber Bragg grating technology has been widely used to provide such custom optical filters. However, in recent years, the fabrication of Bragg gratings in silicon waveguides has become very appealing because CMOS technology provides a low-cost small-footprint platform on which several functions and devices can be integrated. Short and simple grating structures with strong coupling coefficients have already been demonstrated in silicon-on-insulator (SOI) waveguides with good quality spectral responses and integration of such short uniform gratings on a four ports structure has allowed operating the gratings in reflection [1,2]. Furthermore, demonstrations of tunable gratings using the thermo-optic and the electro-optic effects suggests potential use of these devices for modulation and switching [3,4]. However, when elaborate spectral responses are needed long gratings must be used. Unfortunately, straight integrated Bragg gratings (IBG) have an aspect ratio that can make them difficult to integrate efficiently in photonic integrated circuits. Furthermore, long straight gratings will be more affected by non-uniformities of fabrication processes including wafer thickness variations [5,6]. There is therefore a strong interest to make compact grating structures that will provide increased flexibility in the design of photonic integrated circuits on SOI.

Huge research efforts have recently been deployed to minimize losses in curved photonic wires resulting in loss values lower than 0.01 dB/90° for singlemode waveguides with a radius of curvature of 5 μm [7]. Numerical simulations have also shown that hybrid multimode/singlemode waveguides should also exhibit such small bending losses [8,9]. Those advances allow the work presented in this paper, which focuses on the implementation of IBGs in spiral waveguides in order to increase grating length while improving its aspect ratio. When designing such spiral IBGs, the local radius of curvature is relatively large (R > 20 μm in this paper) compared to the radius of curvature considered in [7,9]. With such large radius of curvature, bending losses are not a concern compared to the induced effective index variation. As a result, the analysis of curved waveguides presented in this paper focuses on preventing unwanted variations of the grating phase that would result in spectral distortion.

Several studies have addressed the modeling of bent waveguides. It has been shown that conformal transformations convert circularly curved step index waveguides with a constant radius of curvature and a 1D confinement to simple straight waveguides with modified refractive index profiles [10,11]. Furthermore, the equivalent straight waveguide (ESW) approximation has extended this approach to waveguides having a 2D confinement [12–14]. Later on, it was shown that this approximation is also appropriate for photonic wires with *R* > ~2 μm [15]. In this work, we use this approach to model spiral IBGs by defining an equivalent straight grating that includes an effective index perturbation caused by the waveguide curvature. Then, the spectral response of spiral IBGs can be calculated using a 1-D simulator based on coupled mode equations.

The paper is organized as follows. Firstly, we present the procedure to correct the effective index perturbation caused by the curvature when an IBG is implemented in a spiral waveguide on silicon-on-insulator (SOI). Afterwards, we show experimental results that compare compensated and uncompensated grating structures. The results confirm that the correction was successful and that spiral IBGs can be fabricated without spectral degradation. We further demonstrate a spiral configuration with interleaved waveguides that results in a highly integrated Bragg grating structure. Lastly, we address the apodization of spiral IBGs. When the Bragg grating length is increased, its coupling coefficient is usually reduced and this makes grating apodization more challenging because the modulation of the corrugation amplitude becomes limited by the precision of the fabrication process. However this difficulty can be overcome by using phase apodization as demonstrated in [16]. We show that spiral IBG structures are compatible with phase apodization.

## 2. Waveguide and grating design

To improve IBGs compactness, a zigzag layout has been proposed in which the grating waveguide is bent in a succession of curved sections with uniform radius of curvature thus forming a series of s-shape waveguides [17]. Unfortunately, this approach adds a fair amount of propagation loss due to mode mismatch at the connection point between two curved sections where the center of rotation is moved. Furthermore, at those positions, there is a rapid modification of the effective index, which distorts the grating spectral response. These effects, that increase drastically when the radius of curvature is decreased or when the grating length is increased, strongly jeopardize the merit of this approach in terms of integration capability. Spiral waveguides, as shown in Fig. 1(a), do not induce significant losses since the radius of curvature is on average much larger, except near its center, and there is no discontinuity in the waveguide curvature, which alleviates coupling losses between waveguide sections. As a result, the spiral geometry makes it possible to design long waveguides with large radius of curvature while still improving significantly the compactness of the device. As for the effective index perturbation caused by the variation of the radius of curvature, we present in the remaining of this section an efficient technique to compensate this effect.

As shown in [18], the spectrum of a long IBG is strongly distorted by the effect of sidewall roughness. This effect is enhanced by the high index contrast of SOI waveguides. A means to reduce this phase noise in the design process is to implement the IBG in the multimode section of a hybrid multimode/singlemode waveguide or in a rib waveguide. The first approach was considered in this paper. Singlemode sections with adiabatic couplers are placed before and after the multimode section in order to predominantly excite the fundamental mode of the multimode section. As a result, the grating is behaving as if it were in a singlemode waveguide, but with less distortion. In this work, we have chosen a waveguide having a width (*w*) of 1200 nm since it does not have more than 3 transverse electric (TE) modes and its sensitivity to sidewall roughness is strongly decreased. Input/output light coupling is achieved with compact focusing grating couplers [19]. These couplers are followed by 2 mm long single-TE mode waveguides having a width of 500 nm. Then two 125 μm-long adiabatic tapers connect the single-TE mode waveguides to the multimode waveguide, which contains the 2 mm-long spiral grating. The designed IBG corrugations have sidewall recess amplitude of 30 nm and a mean grating period, Λ, of 858.24 nm. A third order grating was chosen due to the fabrication limitation of a 300 nm minimal pitch. Since the corrugation amplitude is much smaller than the minimum feature size (180 nm), we anticipated that the rectangular corrugations on the CAD mask would be transferred as a sinusoidal function on the silicon layer [20]. Thus, because an index perturbation of a sine shape does not have higher order Fourier term, the grating average duty-cycle, defined as the ratio of the corrugation width to the grating period, has been set to 25% to ensure the presence of a significant third order resonance. The devices were fabricated on a 220 nm thick silicon layer wafer using 248 nm deep UV photolithography [21]. The CAD mask is shown in Fig. 1(b) and an optical microscope image of the first row of the chip is shown in Fig. 1(c). This chip of 600 µm x 1400 µm has been covered by a silica cladding and contains 17 samples of 2-mm long gratings in spiral waveguides surrounded by 2 mm-long waveguides on each side of the IBGs. Since multimode waveguides are more sensitive to curvature losses, the radius of curvature should be larger than about 10 microns. Fortunately, as discussed previously, it is possible to design spiral gratings with small length-footprint ratio without using small radius of curvature.

In order to bend a straight grating into a spiral-shaped grating, three steps need to be taken: 1) a spiral waveguide of the required length is defined, 2) the grating is designed for the equivalent straight waveguides, which means that the grating incorporates the phase compensation term, and 3) the grating is geometrically mapped on the spiral. As mentioned earlier, this procedure uses the ESW approximation to transform the curved waveguide into an equivalent straight waveguide with a modified index profile. We detail below these three steps.

#### 2.1 Spiral waveguide definition

The *x* and *y* coordinates of the spiral are given by the real and imaginary parts of *S* defined by

*ρ*is the angle of rotation that increases along the spiral waveguide (

*ρ*= 0 at the center). The term

*R*(

*ρ*)e

^{i}^{|}

^{ρ}^{|}is the spiral itself with a radius of curvature that is changing linearly with the angle of rotation. Without the Δ

*x*term in Eq. (1), the profile of the radius of curvature would be given by the black dotted line in Fig. 2(a) having a minimum value equal to

*R*at

_{0}*ρ*= 0. A simple spiral could be designed by using an s-shaped waveguide having a radius of curvature of

*R*

_{0}/2 to connect a spiral waveguide defined by

*R*(

*ρ*)e

*, with*

^{iρ}*ρ*> 0, to another one defined by

*R*(

*ρ*)e

*e*

^{iρ}*[22]. However, this leads to a discontinuity in the radius of curvature (jumping from*

^{iπ}*R*to

_{0}*R*

_{0}/2). The addition of the term Δ

*x*in Eq. (1) avoids this discontinuity by smoothly shifting the center of rotation between the two spirals. Figure 2(a) shows the numerically calculated profile of the radius of curvature,

*R(z),*as a function of the position on the spiral path from the input to the output of the waveguide (refer to by “

*z*-position”). When Δ

*x*becomes negligible, outside the red dots in Fig. 1(a) (located in this case at

*ρ = ±*5

*π*/4), the waveguide behaves as a simple spiral with a radius of curvature that increases linearly with

*ρ.*

The profiles of the radius of curvature for three typical spirals are shown by the red, black and blue lines in Fig. 2(a) and the spiral themselves are presented in Fig. 1(a). *R _{0}* = 46.85 µm and Δ

*w*= 12.5 µm for each spirals, while the

*α*parameter, which controls the rate of change of the center of rotation as function of

*ρ*, is respectively 0.3355, 0.671 and 1.0065. On one hand, when

*α*is reduced, the minimum value of the radius of curvature of the spiral becomes smaller and on the other hand, when

*α*is increased, the radius of curvature increases significantly in the central portion of the spiral (around the location [$\mp $40, ± 40] in Fig. 1(a) and at the

*z*-position around ± 0.2 mm in Fig. 2(a)) which decreases the waveguide spacing in this area. Consequently,

*α*must be chosen carefully.

Once the waveguide is defined, the local unitary normal vector, $\overrightarrow{N}$ is found numerically, which allows the calculation of the exact position of the waveguide sidewalls, i.e. $S\pm w\overrightarrow{N}/2$, and eventually allows the calculation of the position of the corrugation amplitudes.

#### 2.2 Grating design

IBGs index profiles are given by

*θ*is a phase term that can be used to incorporate an arbitrary chirp function, Λ is the design grating period,

*n*(

*λ*) is the effective index of the straight waveguide, λ is the optical wavelength, Δ

*n*is the grating index modulation amplitude and δ

*n*(

*R*(

*z*)) is the effective index perturbation caused by the curvature, as shown in Fig. 2(b) for a waveguide of 1200 nm x 220 nm. The dependency of the waveguide effective index as a function of curvature was obtained using a finite elements mode solver simulator combined with the ESW approximation. The calculations take into account the specific profile of the radius of curvature calculated as a function of the

*z*-position for the given spiral. The phase term Ω(

*z*) in Eq. (4) is added to compensate for the effective index distortion

*δn*(

*R*(

*z*)) caused by the spiral and is given by

In this work, we use a third order grating (Λ = 858.24 nm) and design the mask with square corrugations of 25% duty cycle. The unbent position of each rectangular corrugation added on the side of the waveguide is obtained directly from the maximal values of every cycle of the cosine term in Eq. (4). The edge of the corrugation is then simply determined from the duty cycle.

The associated phase correction for the three spirals discussed previously is shown in Fig. 2(c). As can be seen on this figure, when the radius of curvature is large (on the sides), the phase function increases slowly since the effective index perturbation is very small. However, when the radius of curvature becomes smaller, the phase correction varies rapidly and the grating period must be decreased locally in order to compensate for the higher index of refraction. To illustrate the impact of the phase correction term, the grating local period, given by

#### 2.3 Mapping of the grating on the spiral

The last step of the mask design consists of mapping the grating corrugation profile of the straight waveguide onto the path of the spiral shape. The normal coordinate is calculated using $\overrightarrow{N}$and the designed corrugation width. It should be noted that the coupling coefficient asymmetry associated with the gratings on the inner and outer side of a curved waveguide is negligibly small for the radius of curvature considered in this paper (> 20 μm) and therefore, the corrugation amplitude has not been compensated [22].

## 3. Grating characterization

The complex spectral responses of every gratings was measured with a commercial optical frequency domain reflectometer to which we removed the coupling losses. To eliminate the strong reflections at the end of the cleaved fibers, we performed time filtering of the grating impulse response. The 2 mm-long single-mode waveguide sections on both ends of the spiral provided sufficient temporal separation to filter out the unwanted reflections. The experimental measurements are shown in red in the following figures presenting experimental results (Fig. 3(a), Fig. 4(a), Fig. 5(d) and Fig. 6(a)). We then retrieved the Bragg wavelength (*λ _{B}*(

*z*)), which is equal to 2

*n*(

*λ*)Λ(z), and the index modulation amplitude profiles, Δ

*n*(

*z*), of the IBGs using an inverse scattering algorithm, namely the integral layer peeling algorithm proposed in [23]. The maximal grating reflectivity was designed to be small to ease the convergence of the grating reconstruction algorithm. However, it should be mentioned that spiral-IBGs are not limited to weak gratings. The corrugation recess amplitude can be increased as easily as for straight IBGs. The retrieved grating amplitude and phase profiles, after appropriate filtering, are used to calculate the reconstructed spectral responses using a standard transfer matrix solution of the coupled mode equations (black curves in Fig. 3(a), Fig. 4(a), Fig. 5(d) and Fig. 6(a)). The good correspondence between the reconstructed spectra and the measured ones indicate that the

*λ*and Δ

_{B}*n*profiles were retrieved with sufficient precision. More details on the post-processing procedure can be found in [24]. The designed grating spectral responses are shown in blue in Fig. 3(a), Fig. 4(a), Fig. 5(d) and Fig. 6(a). The grating spectra displayed in this paper are typical results. Over thirty gratings have been characterized with similar responses.

## 4. Experimental results and discussion

#### 4.1 Phase correction effect

We designed two uniform IBGs on a spiral having the following parameters: *R*_{0} = 46.85 µm, Δ*w* = 12.5 µm and *α* = 0.671 which results in spirals having a minimal radius of curvature of 20 µm, a length of 2 mm and an area of 200 µm x 190 µm. The spirals have the same parameters as those shown in Fig. 1(a) with *α* = 0.671.

The first grating, presented in Fig. 3, was made without phase correction while the second grating, presented in Fig. 4, was designed with the phase correction. The spectral response of both measured and reconstructed gratings are shown in Fig. 3(a) and Fig. 4(a) while their retrieved physical profiles are shown in Fig. 3(b) and Fig. 3(c) and Fig. 4(b) and Fig. 4(c). As expected, the Bragg wavelength profile of the uncorrected grating shows a strong Bragg wavelength perturbation near its center. The spiral curvature acts as a distributed phase shift which results in a strong resonance in the grating bandgap as shown in Fig. 3(a). In this case, the distributed phase shift seems to be reasonably close to π since the resonance in the grating band is strong and centered in the grating band, which might be of interest for some applications such as notch filters. Simulations of discrete phase-shifted gratings actually show that this asymmetry between the two main lobes can be obtained with a centered phase-shift of ~1.15π. However, for most applications, the grating effective index should be corrected in order to prevent spectral distortions. Figure 3(b) shows that the Bragg wavelength perturbation is cancelled out in the corrected grating displayed in Fig. 4(b) showing that the phase compensation approach is working properly. The grating sidelobe suppression ratio shown in Fig. 4 (~10 dB) is slightly better than what has already been published for identical 2 mm-long straight grating [16]. Because the spiral grating is localized on a smaller area of the wafer, the phase noise created by silicon thickness variations is less likely to have an impact on the grating spectral response compared to gratings in straight waveguides.

Finally, it might seem odd that the phase shift in the Bragg wavelength profile of Fig. 3(b) does not experience a strong minimum at its center since the spiral is straight at this particular section of the grating and hence it should not experience phase distortion. However, Bragg gratings are not affected by the high frequency spatial components of the phase and index modulation profiles [18,24]. Furthermore, only the low frequency components can be measured. As a result, the Bragg grating reconstruction acts as a low-pass filter which suppresses the localized minimum inside the spiral-induced distortion.

#### 4.2 Highly integrated gratings

In order to improve the grating integration factor, we designed a spiral structure consisting of three interleaved gratings. The addition of two other spirals (in red and blue in Fig. 5(a)) beside the central spiral (in black) is straightforward since the spiral function (*S*) and $\overrightarrow{N}$ arealready known. As a result, the center of the side-spirals waveguides are defined by ${S}_{\pm}=S\pm (\Delta w/3)\overrightarrow{N}$. Aside from this detail, the design of those side-spiral gratings is exactly the same as discussed above. In this work, we chose spirals with *R*_{0} = 59 µm, Δ*w* = 15 µm, *α* = 0.671 and *N* = 1.345 which gives three spirals spaced by 5 µm, having a minimal radius of curvature of 20 µm for the side-spirals and 25 µm for the central spiral, a path length of 2 mm (each spiral has the same total length) and a total area of 230 µm x 215 µm. It can be noticed that the total footprint of these three IBGs is almost the same as the single spiral grating described in the previous section, thereby tripling the integration efficiency. The radius of curvature of the three gratings and their associated phase correction terms are shown in Fig. 5(b) and Fig. 5(c) respectively. As can be seen in Fig. 5(b), the central spiral has a symmetric radius of curvature profile. However, even if the side spirals experienced asymmetric profiles, their physical structure can be as easily compensated. Figure 5(d) shows that the interleaved spiral does not affect significantly the grating spectrum as well as the fact that each waveguides are much closer than in the previous section. As it was the case for single-spiral, the correction phase term compensates well the effective index perturbation caused by the curvature; at least, the phase error has amplitude in the center portion of the grating that is smaller than the random phase noise observed near its input and output ends.

#### 4.3 Apodized gratings

Finally, since a full control of the complex spectral responses of IBGs requires reliable apodization techniques, the results presented in this last section confirms that spiral waveguides are compatible with the phase-apodization technique presented in [16]. Briefly, this technique adds a slow phase modulation in the grating structure with a *z*-dependent amplitude (*ϕ*(*z*)). Consequently, the last term of Eq. (4) representing the grating can be written in the form

Phase modulation at a spatial frequency 1/Λ* _{m}* results in amplitude apodized grating with a spectral response having satellite resonances out of the band of interest when Λ

*is sufficiently small (in the present case Λ*

_{m}*= 17.5 μm). Figure 6(c) shows that the effective index modulation of the grating now follows the designed Gaussian profile with a 1 mm full-width half-maximum indicating that the correction applied to the grating does not affect the apodization profile. Unfortunately, IBGs in SOI usually experience a fair amount of phase noise, which prevents the apodization to properly reduce the sidelobe amplitude. To confirm this point, the green curve of Fig. 6(a) presents the spectral response obtained using the grating apodization profile shown in Fig. 6(c), but with an ideal Bragg wavelength profile (no phase noise). As a result, the sidelobes suppression ratio is decreased by more than 20 dB, which corresponds closely to the design (in blue). Consequently, the variation of the apodization profile compared to the ideal design is not a major source of spectral distortion and we thus conclude that spiral IBGs are compatible with phase-apodization.*

_{m}## 5. Conclusion

We presented a simple procedure to design IBGs in a spiral configuration to improve their integration. We showed that IBGs in spiral waveguide can be fabricated without additional phase-noise or spectral distortions due to waveguide curvature. We further proposed an interleaved spiral configuration that allows fabrication of many gratings on the same chip section, thereby multiplying the integration factor. Finally, we have confirmed that this work is compatible with the phase-apodization technique needed for the fabrication of grating-based devices with elaborate spectral responses. Future work should consider the optimization of the waveguide spacing, Δ*w*, in order to improve the compactness of the spiral-gratings. This work shows that as the quality of SOI wafers improves, long IBGs with high quality spectral characteristics will be achievable. This design approach can be used for various grating types and strengths thereby giving increased flexibility for the layout of photonic circuits.

## Acknowledgments

This research was funded by the Canada research chair APTECS, NSERC and CMC Microsystems. The authors acknowledge Dr. Dan Deptuck and Mr. Nicolas Ayotte for insightful discussions. The chips described in this paper were fabricated using the OpSIS service through IME A*STAR in Singapore.

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