Abstract

Spectral filters are important building blocks for many applications in integrated photonics, including datacom and telecom, optical signal processing and astrophotonics. Sidewall-corrugated waveguide grating is typically the preferred option to implement spectral filters in integrated photonic devices. However, in the high-index contrast silicon-on-insulator (SOI) platform, designs with corrugation sizes of only a few tens of nanometers are often required, which hinders their fabrication. In this work, we propose a novel geometry to design complex Bragg filters with an arbitrary spectral response in silicon waveguides with laterally coupled Bragg loading segments. The waveguide core is designed to operate with a delocalized mode field, which helps reduce sensitivity to fabrication errors and increase accuracy on synthesized coupling coefficients and the corresponding spectral shape control. We present an efficient design strategy, based on the layer-peeling and layer-adding algorithms, that allows to readily synthesize an arbitrary target spectrum for our cladding-modulated Bragg gratings. The proposed filter concept and design methodology are validated by designing and experimentally demonstrating a complex spectral filter in an SOI platform, with 20 non-uniformly spaced spectral notches with a 3-dB linewidth as small as 210 pm.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Bragg gratings are structures with a periodically modulated refractive index profile along the propagation direction of light with a period approximately one half of the effective wavelength [1]. They reflect a specific spectral range of the incoming light while transmitting the other wavelength components, thus allowing spectral filtering. Although Bragg gratings have been mostly implemented in optical fibers [13], many efforts have also been directed to developing Bragg gratings in planar waveguides, including silicon-on-insulator and silicon nitride platforms [435].

Bragg gratings can be efficiently used for flexible spectral tailoring. It has been shown that a filter with an arbitrary spectral response can be designed by modulating the coupling coefficient $\kappa $ (the grating strength) along the grating [36,37]. This technique, also known as apodization, has enabled the design of filters with flexible spectral features for applications such as wavelength-division multiplexing (WDM) [11,12,18,27,28], optical signal processing [13,15,17], optical communications [6,10,20,31], or astrophotonics [33,35], to name a few.

Bragg filters in integrated photonic platforms have been typically implemented with sidewall-corrugated gratings, i.e. by modulating the waveguide width. In practice, this method is limited by the minimum corrugation size allowed in the fabrication process, which determines the weakest perturbation that can be achieved and, in turn, the minimum coupling coefficient $\kappa $ and filter bandwidth. This constraint becomes especially relevant in the SOI platform, where the filter performance is highly sensitive to small variations of the corrugation width because of the high refractive index contrast. For example, waveguide sidewall corrugations of 10 nm or less, hence challenging to fabricate, are required to achieve sub-nanometer bandwidths in a 220-nm-thick SOI platform [4]. Advanced modulation schemes have been proposed to ease this constraint, including misaligned sidewall corrugations [5,9], grating pitch modulation [5] or subwavelength-engineered sidewall gratings [2123]. Photonic Hilbert transformers [13,15] and multi-channel filters [1820] have been demonstrated by using these techniques. However, corrugation widths quite challenging to reproducibly fabricate (< 20 nm) are typically required [13,15,1820].

Cladding-modulated Bragg gratings, i.e. structures with the periodic perturbation physically separated from the waveguide core, are an interesting alternative to implement spectral filters in silicon waveguides. Specifically, periodic arrays of lateral cylinders [24,25] and sidewall corrugations on lateral silicon strips parallel to the waveguide [26] have been proposed. These geometries allow to achieve weak Bragg gratings with narrow spectral features by judiciously positioning the grating in proximity of the waveguide core. At the same time, designs with relaxed minimum feature sizes can be obtained. Gaussian-apodized gratings [27] and multi-band filters [28] have been demonstrated based on cladding-modulated gratings.

Due to the strong mode confinement in silicon waveguides, in cladding-modulated gratings it is challenging to accurately control the coupling coefficient by adjusting the separation between the waveguide core and the Bragg grating. At the same time, the maximum achievable coupling coefficient is limited by the minimum separation between the waveguide core and the grating allowed by the fabrication constraints. These limitations can be potentially circumvented by reducing the effective index contrast, hence delocalizing the waveguide mode. In this paper, we investigate cladding-modulated gratings operating with TM polarized light, which has a reduced confinement compared to TE. Then, we analyze two different strategies to implement the waveguide core with enhanced mode delocalization, namely i) a homogeneous silicon wire, designed with a reduced waveguide width, and ii) a subwavelength metamaterial core with a reduced equivalent refractive index. Subwavelength gratings (SWGs), since their early demonstrations in silicon waveguides [3843], have become established as a fundamental tool to control local electromagnetic field distribution in integrated photonic devices [4447]. Notch filters with bandwidths ranging from 150 pm to 8 nm and a minimum feature size as large as 100 nm have been designed [29] and experimentally demonstrated [30], leveraging an SWG core laterally loaded with a periodic array of silicon segments. More recently, a similar strategy has also been implemented in add-drop filters [31] and optical delay lines [32]. These advances have opened a promising path towards the development of advanced waveguide filters with complex spectral features and relaxed fabrication tolerances in silicon waveguides.

Here we investigate cladding-modulated Bragg gratings in silicon waveguides with reduced mode confinement, to design filters with a complex spectral response. In our filters, the grating coupling coefficient is controlled by modulating the distance between the waveguide core and the loading segments, as shown in Fig. 1. We develop an efficient design methodology based on the layer-peeling (LPA) [36,37] and layer-adding (LAA) [33] algorithms, optimized for silicon waveguides. We examine in detail the advantages of using the proposed waveguide geometries with reduced mode confinement to implement spectral filters with an arbitrary spectral response. As a proof of concept, we design and experimentally demonstrate a complex filter that synthesizes a transmittance spectrum comprising 20 non-uniformly spaced notches, relevant for application in OH emission lines suppression in astronomical observations [33].

 figure: Fig. 1.

Fig. 1. Schematic view of the cladding-modulated Bragg filter geometries analyzed in this paper: (a) continuous silicon waveguide core and (b) SWG metamaterial waveguide core (SiO2 upper cladding is not shown for clarity). Filter spectral response is controlled by modulating the separation (${s_n}$) between the waveguide core and the loading segments and the grating period (${\mathrm{\Lambda }_n}$). (c-d) Scanning electron microscope (SEM) images of a short section of the fabricated filters. A 2.2-µm upper SiO2 cladding was deposited by plasma-enhanced chemical vapor deposition after the SEM images were taken.

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This article is organized as follows. In section 2, the proposed Bragg filter geometries are discussed. In section 3, we present the design strategy to synthesize filter spectral response. Simulation and experimental results are presented in sections 4 and 5. The conclusions are drawn in section 6.

2. Filter geometry

We study two geometries of cladding-modulated Bragg filters, with i) a silicon wire core (width W, see Fig. 1(a)) and ii) an SWG waveguide core (period ${\mathrm{\Lambda }_{\textrm{SWG}}}$, duty cycle $\textrm{D}{\textrm{C}_{\textrm{SWG}}} = a/{\mathrm{\Lambda }_{\textrm{SWG}}}$ and width W, see Fig. 1(b)). In both cases, the Bragg grating is formed by loading segments with length ${L_\textrm{S}}$ and width ${W_\textrm{S}}$, laterally separated from the core and modulating in a controlled manner the evanescent field of the propagating mode. Compared to other commonly used geometries for sidewall corrugated gratings [423], this structure allows to accurately synthesize low coupling coefficients, as required for narrowband filters. At the same time, it permits using comparatively large segments and placing them sufficiently far from the waveguide core, relaxing the minimum feature size requirements. To synthesize a target spectral response, we modulate both the separation between the waveguide core and the loading segments, ${s_n}$, and the Bragg pitch, ${\mathrm{\Lambda }_n}$, for the $n$-th Bragg period. The separation modulation ${s_n}$ along the grating allows to synthesize the required Bragg coupling coefficient profile ${\kappa _n}$ for the target frequency response of the filter. The small adjustment of Bragg period ${\mathrm{\Lambda }_n}$ allows to synthesize the phase profile required for the impulse response of the filter, as well as to compensate for small variations of the effective index caused by the presence of the lateral loading segments. The ${s_n}$ and ${\mathrm{\Lambda }_n}$ are the key parameters determining the filter response and are calculated as outlined in the following section.

3. Design methodology

We followed the following methodology to determine the ${s_n}$ and ${\mathrm{\Lambda }_n}$ that synthesize a target reflection spectrum $r(\lambda )$. First, we design a simple (non-apodized) filter geometry, i.e., the waveguide core and the dimensions of the Bragg loading segments, and we calculate the group index of the unperturbed waveguide (${n_{\textrm{g},\textrm{u}}}$) and the grating coupling coefficients ($\kappa $ and $\kappa ^{\prime}$) as a function of the separation s of the loading segments. In the second step, we determine the local reflection coefficients to be implemented along the filter, $\; {\tilde{\rho }_n}$ (see Figs. 1(a) and 1(b)). These coefficients are obtained as follows: i) from the targeted spectral response $r(\lambda )$, the ideal reflection coefficients ${\rho _n}$ are obtained by means of the LPA [36,37]; ii) the ideal reflection coefficients are truncated, for filter length reduction, thus obtaining a shorter coefficient set ${\tilde{\rho }_n}$ which can be practically implemented; iii) the LAA [33] is applied to obtain the expected spectral response of the filter $\tilde{r}(\lambda )$. Finally, we map the ${\tilde{\rho }_n}$ distribution onto the coupling coefficient profile ${\kappa _n}$, and calculate the lateral separation ${s_n}$ and Bragg period ${\mathrm{\Lambda }_n}$ distributions along the filter.

The different steps of our design methodology are schematically shown in Fig. 2 and are discussed in detail in the following subsections.

 figure: Fig. 2.

Fig. 2. Design flow to implement filters with an arbitrary frequency response. Part 1: Determination of the basic filter geometry and coupled-mode parameters. The waveguide core parameters are designed to achieve a low effective index and a correspondingly increased mode delocalization, and fulfill the Bragg condition at the central wavelength ${\lambda _0}$. Then we calculate the group index ${n_{\textrm{g},\textrm{u}}}({{\lambda_0}} )$, and the coupling $\kappa (s )$ and self-coupling $\kappa ^{\prime}(s )$ coefficients of the grating as a function of the separation s of the loading segments. Part 2: Calculation of the local reflection coefficients by using layer-peeling and layer-adding algorithms. The target spectrum in the wavelength domain, $r(\lambda )$, is first converted into the wavenumber domain, $r(\delta )$. The LPA returns the ideal local reflection coefficient profile ${\rho _n}$. This profile is later shortened to minimize the filter length, then obtaining the final reflection coefficient profile ${\tilde{\rho }_n}$ to be implemented along the filter. The corresponding spectral response $\tilde{r}(\lambda )$ is calculated by using the LAA. Part 3: Filter apodization. The local reflectivity profile ${\tilde{\rho }_n}$ is synthesized period by period. The absolute value $|{{{\tilde{\rho }}_n}} |$ is implemented with the separation ${s_n}$ by using Eq. (8) and the mapping function $\kappa (s )$. The phase $\angle {\tilde{\rho }_n}$ is synthesized by adjusting the period ${\mathrm{\Lambda }_n}$ according to Eq. (10) and the mapping function $\kappa ^{\prime}(s )$.

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3.1 Design of the basic filter geometry and calculation of coupled-mode parameters

Here we discuss the electromagnetic analysis and design of the non-apodized filter geometry, namely a periodic structure with constant separation s and constant Bragg period ${\mathrm{\Lambda }_\textrm{B}}$, as schematically outlined in Fig. 2, part 1. We assume the following considerations:

  • a) Modal effective index and field delocalization is controlled through the waveguide width W. The Bragg period is then calculated to fulfill the Bragg condition for the unperturbed waveguide core at the central wavelength ${\lambda _0}$:
    $${\mathrm{\Lambda }_\textrm{B}} = \frac{{{\lambda _0}}}{{2{n_{\textrm{eff},\textrm{u}}}({{\lambda_0}} )}}.$$

    For our design with SWG waveguide core, two additional parameters need to be specified: the duty cycle $\textrm{D}{\textrm{C}_{\textrm{SWG}}}$ and the SWG period ${\mathrm{\Lambda }_{\textrm{SWG}}}$. The $\textrm{D}{\textrm{C}_{\textrm{SWG}}}$ allows to accurately control the effective index and, along with the SWG segment width W, is chosen to maximize the field delocalization while avoiding substrate leakage losses [48]. The grating pitch ${\mathrm{\Lambda }_{\textrm{SWG}}}$ needs to be selected to operate sufficiently far away from the Bragg regime. We choose ${\mathrm{\Lambda }_{\textrm{SWG}}} = {\mathrm{\Lambda }_\textrm{B}}/2$ [29,30].

  • b) The dimensions of the Bragg loading segments (${L_\textrm{S}}$ and ${W_\textrm{S}}$) are selected large enough to meet the minimum feature size requirements (> 100 nm).

Then, the relevant coupled mode parameters of the structure are calculated, specifically the group index of the unperturbed waveguide at the central wavelength, ${n_{\textrm{g},\textrm{u}}}({{\lambda_0}} )$, and the coupling and self-coupling coefficients of the Bragg grating as a function of the separation of the loading segments, $\kappa (s )$ and $\kappa ^{\prime}(s )$ [29]. The latter will be used in the subsequent steps to map the reflection coefficient profile $\; {\tilde{\rho }_n}$ into the filter geometrical parameters ${s_n}$ and ${\mathrm{\Lambda }_n}$.

3.2 Calculation of the local reflection coefficients: layer-peeling and layer-adding algorithms

Our implementation of the LPA is based on the discrete LPA model described in detail in [37]. This algorithm allows the apodization profile to be synthesized in terms of the ideal, complex, local reflection coefficients between adjacent Bragg periods, ${\rho _n}$. The amplitude and phase of the local reflection coefficients are denoted as $|{{\rho_n}} |$ and $\angle {\rho _n}$, respectively.

To determine these coefficients, we start with the target reflection spectrum $r(\lambda )$, fulfilling the causality condition, in order to be physically realizable [15,37]. The LPA equations are formulated in terms of the wavenumber detuning $\delta $ with respect to the Bragg resonance. The wavelength is mapped into wavenumber detuning through the central wavelength ${\lambda _0}$ and the group index ${n_{\textrm{g},\textrm{u}}}$ [49]:

$$\delta (\lambda )= {\beta _\textrm{u}}(\lambda )- \frac{{{K_\textrm{B}}}}{2} \approx{-} 2\mathrm{\pi }{n_{\textrm{g},\textrm{u}}}({{\lambda_0}} )\frac{{\lambda - {\lambda _0}}}{{\lambda _0^2}},$$
where ${\beta _\textrm{u}}$ is the propagation constant of the unperturbed waveguide and ${K_\textrm{B}} = 2\mathrm{\pi }/{\mathrm{\Lambda }_\textrm{B}}$ is the grating constant. Equation (2) is used to map the target spectrum given in the wavelength domain, $r(\lambda )$, into a target spectrum defined with respect to the wavenumber detuning, $r(\delta )$, required in the subsequent LPA step (see Fig. 2, part 2).

The LPA takes as input the reflection spectrum at the position of the $n$-th Bragg period, ${r_n}(\delta )$ (see Figs. 1(a) and 1(b)). The local reflection coefficient ${\rho _n}$ is obtained by applying the causality argument on the impulse response of ${r_n}(\delta )$, i.e., evaluating the zeroth coefficient of its discrete time inverse Fourier transform [37]:

$${\rho _n} = \frac{{{\mathrm{\Lambda }_\textrm{B}}}}{\mathrm{\pi }}\mathop \int \nolimits_{ - \mathrm{\pi }/2{\mathrm{\Lambda }_\textrm{B}}}^{\mathrm{\pi }/2{\mathrm{\Lambda }_\textrm{B}}} {r_n}(\delta )\; \textrm{d}\delta .$$

The effect of the $n$-th period is then removed and the reflection spectrum at the next position, ${r_{n + 1}}(\delta )$, is calculated as [37]:

$${r_{n + 1}}(\delta )= \textrm{exp} ({ - \textrm{j}2\delta {\mathrm{\Lambda }_\textrm{B}}} )\frac{{{r_n}(\delta )- {\rho _n}}}{{1 - \rho _n^\ast {r_n}(\delta )}}.$$

By defining the reflection spectrum of the first period as the target reflection spectrum (${r_1}(\delta )= r(\delta )$), iterative application of Eqs. (3) and (4) for growing values of n provides the local reflection coefficients ρn of the full structure (see Fig. 2, part 2).

The LPA can be applied from $n = 1\; $ to an arbitrary large value of n so that all significant information of the impulse response is included. This usually leads to large number of coefficients and, therefore, long filters. In most situations, the leading and trailing edges of the ${\rho _n}$-profile have little influence on the synthesized spectrum [16]. This fact can be advantageously used to reduce the length of the filter by truncating it for $n \in [{{N_1},{N_2}} ]$, i.e., implementing a shortened local reflection coefficient profile ${\tilde{\rho }_n}$ defined as:

$${\tilde{\rho }_n} = {\rho _{n + {N_1} - 1}}\; ,\; \; \; 1 \le n \le \tilde{N}$$
where $\tilde{N} = {N_2} - {N_1} + 1$ is the number of Bragg periods of the shortened filter. The spectrum synthesized by this filter can be simulated using the LAA, which is derived from the LPA and is based on the iterative application of the following expression [33]:
$${\tilde{r}_n}(\delta )= \frac{{{{\tilde{r}}_{n + 1}}(\delta )+ {{\tilde{\rho }}_n}\textrm{exp} ({ - \textrm{j}2\delta {\mathrm{\Lambda }_\textrm{B}}} )}}{{\textrm{exp} ({ - \textrm{j}2\delta {\mathrm{\Lambda }_\textrm{B}}} )+ \tilde{\rho }_n^\ast {{\tilde{r}}_{n + 1}}(\delta )}},\; \; \; \tilde{N} - 1 \ge n \ge 1.$$

When ${N_1}$ and ${N_2}$ are properly chosen, the synthesized spectrum $\tilde{r}(\delta )= {\tilde{r}_1}(\delta )$ calculated from Eq. (6) closely resembles the target spectrum $r(\delta )$, but with a significant reduction of the filter length. This procedure is schematically outlined in Fig. 2, part 2.

Note that in Eqs. (4) and (6), the spatial dependence of the fields is assumed to be of the form $\textrm{exp} ({ + \textrm{j}\beta z} )$ for forward-propagating modes and $\textrm{exp} ({ - \textrm{j}\beta z} )$ for backward-propagating modes, as in [32,37].

3.3 Filter apodization

From the calculated local reflection coefficients ${\tilde{\rho }_n}$, the coupling coefficient ${\kappa _n}$ for each Bragg period is determined from the coupled-mode theory [37]:

$${\kappa _n} = \frac{{\textrm{tan}{\textrm{h}^{ - 1}}({|{{{\tilde{\rho }}_n}} |} )}}{{{\mathrm{\Lambda }_\textrm{B}}}}.$$

The separation of the loading segments ${s_n}$ is then directly calculated by using the coupling coefficient mapping $\kappa (s )$ that was obtained in part 1 of the design process (see Fig. 2).

To implement the required local reflection coefficient phase $\angle {\tilde{\rho }_n}$, the Bragg periods ${\mathrm{\Lambda }_n}$ need to be adjusted. To this end, we enforce the local reflections produced at the discontinuities n and $n + 1$ (as shown in Figs. 1(a) and 1(b)) to interfere with the correct phase difference at the central wavelength ${\lambda _0}$:

$$2{\mathrm{\Lambda }_n}{\beta _n}({{\lambda_0}} )= 2\mathrm{\pi } + \angle {\tilde{\rho }_{n + 1}} - \angle {\tilde{\rho }_n},$$
which corresponds to the phase condition of the round-trip propagation delay at the Bragg period n. The propagation constant of the n-th Bragg period (${\beta _n}$) at the wavelength ${\lambda _0}$ is related to the grating self-coupling coefficient [28]:
$${\beta _n}({{\lambda_0}} )= {\beta _\textrm{u}}({{\lambda_0}} )+ \kappa _n^{\prime} = \frac{\mathrm{\pi }}{{{\mathrm{\Lambda }_\textrm{B}}}} + \kappa _n^{\prime}.$$

From Eqs. (8) and (9), it follows that:

$${\mathrm{\Lambda }_n} = {\mathrm{\Lambda }_\textrm{B}}\left( {\frac{{2\mathrm{\pi } + \angle {{\tilde{\rho }}_{n + 1}} - \angle {{\tilde{\rho }}_n}}}{{2\mathrm{\pi } + 2{\mathrm{\Lambda }_\textrm{B}}\kappa_n^{\prime}}}} \right).$$

This equation allows to find the modulated Bragg period ${\mathrm{\Lambda }_n}$ from the reference Bragg period ${\mathrm{\Lambda }_\textrm{B}}$, the phase relation between neighboring local reflections $\angle {\tilde{\rho }_{n + 1}} - \angle {\tilde{\rho }_n}$, and the self-coupling coefficient $\kappa _n^{\prime}$. The latter is obtained from the separation ${s_n}$ using the function $\kappa ^{\prime}(s )$ determined in part 1. This part of the design process is outlined in Fig. 2, part 3.

4. Design example: OH emission line suppression filter

In this section, we use the filter topologies investigated in the previous sections to synthesize a complex transmittance spectrum comprising 20 narrow notches at specific wavelengths corresponding to OH emission lines near 1550 nm (see the filter specification in Fig. 3). This type of filter is relevant for ground-based astronomical observations, specifically to suppress strong OH radical emission lines generated in the upper atmosphere [33].

 figure: Fig. 3.

Fig. 3. The filter specification: (a) target transmittance spectrum and (b) design parameters.

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We will compare the characteristics of the two topologies (Fig. 1(a) and Fig. 1(b)), highlighting the potential advantages of each one. The filters are designed for TM polarization, providing an increased mode delocalization compared to TE polarization. This helps to relax the fabrication constraints by increasing the separations of the lateral loading segments as well as reducing the phase de-coherence for long filters, since the effect of fabrication imperfections is reduced. The advantages of using TM polarization for implementing narrow-line Bragg gratings in SWG engineered waveguides have recently been demonstrated in [30].

Our first filter design is based on an Si-wire waveguide of width ${W} = $ 400 nm, while the second design uses an SWG core of width $W = $ 450 nm, pitch ${\mathrm{\Lambda }_{\textrm{SWG}}} = $ 255 nm and duty cycle $\textrm{D}{\textrm{C}_{\textrm{SWG}}} = $ 60%. These SWG parameters were chosen to provide the maximum mode delocalization while keeping substrate leakage loss penalty below 0.7 dB/cm, for a 3-µm-thick buried oxide (BOX) layer [48]. The underlying SWG design strategy is that greater mode delocalization will allow larger loading segments size and an increased separation from the core, relaxing fabrication constraints. Figure 4 shows the effective index ${n_{\textrm{eff},\textrm{u}}}(\lambda )$ and group index ${n_{\textrm{g},\textrm{u}}}(\lambda )$ of the unperturbed waveguides, obtained by our in-house 3D simulation tool based on the Fourier modal method (FMM) [50]. It is observed that the group index of the Si-wire waveguide is large compared to the SWG waveguide, which is expected to result in shorter filter lengths, since the time-to-space mapping $z = t \cdot c/{n_\textrm{g}}$ is related to the group index. On the other hand, the Si-wire design exhibits greater wavelength dependence on the group index, which is expected to slightly distort the filter wavelength response due to non-linear time-to-space mapping.

 figure: Fig. 4.

Fig. 4. (a) Effective index and (b) group index of the unperturbed Si-wire and SWG waveguides as a function of wavelength.

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The grating period ${\mathrm{\Lambda }_\textrm{B}}$ is calculated from the Bragg condition (Eq. (1)) to achieve a central wavelength ${\lambda _0}$ of the target spectrum. For the SWG-based design, we introduce an additional condition, ${\mathrm{\Lambda }_\textrm{B}} = 2{\mathrm{\Lambda }_{\textrm{SWG}}}$, as discussed in section 3. With these considerations, the Bragg periods for the Si-wire and SWG waveguides are $\mathrm{\Lambda }_\textrm{B}^{\textrm{Si} - \textrm{wire}} = $ 458 nm and $\mathrm{\Lambda }_\textrm{B}^{\textrm{SWG}} = $ 510 nm. The loading segments size is chosen large enough to ease fabrication while still maintaining a comparatively small perturbation of the waveguide mode. Table 1 summarizes the main dimensional parameters for our two designs.

Tables Icon

Table 1. Dimensional parameters of the Si-wire and SWG based filter designs.

By applying the LPA on the target spectral function shown in Fig. 3 and then using Eq. (7), we obtain the coupling coefficient distributions ${\kappa _n}$ shown in Fig. 5. As discussed in section 3, the tails with negligible contribution have been truncated to reduce the device length [16]. We have verified that the LAA-responses are indistinguishable from the original target, making sure that the truncation does not practically affect the filter response. The total filter length is ${L^{\textrm{Si} - \textrm{wire}}} \cong $ 3.6 mm for the Si-wire-core design and ${L^{\textrm{SWG}}} \cong $ 6.4 mm for the SWG-core design. The length difference is due to different group indexes for the two designs. It is observed that the longer length for the SWG-based filter results in a weaker coupling coefficient, while the product $L\cdot{\kappa _{\textrm{max}}}$ is maintained constant in both designs.

 figure: Fig. 5.

Fig. 5. Coupling coefficient distributions for (a) Si-wire and (b) SWG based filter designs.

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The final step in our design is the mapping of the coupling coefficient distribution ${\kappa _n}$ onto the loading segment separation ${s_n}$ and the grating period distribution ${\mathrm{\Lambda }_n}$. This stage requires the previous calculation of the coupling and the self-coupling coefficients ($\kappa $ and $\kappa ^{\prime}$ respectively) as a function of the loading segments separation s using an electromagnetic simulator [29]. Figure 6 shows the calculated coupling and self-coupling coefficients for the Si-wire and SWG designs. As expected, $\kappa (s )$ and $\kappa ^{\prime}(s )$ exhibit an exponential dependence on separation s (note that the vertical axis is in a logarithmic scale) because of the exponential decay of the modal field in the lateral cladding region, where the loading segments are placed. On the other hand, the effect of the reduced mode confinement in the SWG design is also obvious, since the separation distance, for any specific coupling coefficient, is increased for this design. Furthermore, the slope of the curve is reduced for the SWG design, relaxing requirements on the positioning accuracy of the loading segments.

 figure: Fig. 6.

Fig. 6. (a) Coupling and (b) self-coupling coefficients as a function of the separation s between the waveguide core and the loading segments, for the Si-wire and SWG waveguides.

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Using data shown in Fig. 6 as a lookup table, we determine the separation profile ${s_n}\; $ and the period distribution ${\mathrm{\Lambda }_n}\; $ shown in Fig. 7. The minimum gap required to implement the maximum coupling coefficient is 141 nm for the Si-wire filter and 392 nm for the SWG filter, so both structures are compatible with deep-UV lithography, which typically requires a minimum feature size on the order of 100 nm. As can be seen in Fig. 8, the separation distribution ${s_n}\; $ is substantially broader for the SWG design compared to the Si-wire filter, easing positional accuracy requirements when placing the segments.

 figure: Fig. 7.

Fig. 7. Final filter design: loading segment separation profile ${s_n}$ for the (a) Si-wire and (b) SWG filters; grating period profile ${\mathrm{\Lambda }_n}$ for the (c) Si-wire and (d) SWG filters.

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 figure: Fig. 8.

Fig. 8. Final filter design: relative frequency distribution of the separations of the loading segments for the Si-wire (blue) and SWG (red) filters. Both histograms have been calculated using the same number of intervals.

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5. Experimental results

We have fabricated the filters on a 220 nm SOI platform with a 2 µm BOX layer. The devices were patterned by electron beam lithography and the inductively coupled plasma - reactive ion etching (ICP-RIE) process [51]. After the device patterning, a 2.2 µm upper cladding SiO2 layer was deposited by a plasma-enhanced chemical vapour deposition (PECVD) process. Finally, the chip facets were defined with a deep-etch process. Figures 1(c) and 1(d) show SEM images (taken before the SiO2 cladding deposition) of a section of two of the fabricated filters. To measure the filter spectral transmittance, light from a tunable laser source was coupled to a polarization controller to select TM polarization, and subsequently to the chip using a lensed SMF-28 optical fiber and an integrated edge coupler. The latter is a high-efficiency broadband SWG metamaterial fiber-chip coupler [5254], connected to the filter via a 500-nm-wide interconnecting waveguide. The light exits the chip through another SWG coupler at the opposite chip edge, being intercepted by a microscope objective, filtered by a Glan-Thompson prism polarizer, and collected by a photodiode connected to a power meter.

 Figure 9 shows the measured transmittance for the Si-wire and SWG waveguide filters. All 20 peaks of the original target spectrum are clearly resolved. There is a noticeable shift of the central wavelength towards longer wavelengths, $\mathrm{\Delta }{\lambda ^{\textrm{Si} - \textrm{wire}}}\; \sim $ 37 nm for the Si-wire filter and $\mathrm{\Delta }{\lambda ^{\textrm{SWG}}}\; \sim $ 31 nm for the SWG filter. We have determined by simulations that this wavelength shift can be explained considering a deviation in the silicon layer thickness $\mathrm{\Delta }H\; \sim $ 10 nm and the size of the silicon segments $\mathrm{\Delta }W\; \sim $ 20 nm, which is within the tolerances of our fabrication process. It is also observed that the SWG filter loss is significantly larger compared to the Si-wire filter, while for the latter the loss penalty is practically negligible. This extra loss is attributed to substrate leakage losses, since the filter, originally designed for a 3 µm BOX, was eventually fabricated on SOI with a 2 µm BOX (3 µm BOX was not available at the time). This assumption is also corroborated by comparing simulated substrate leakage losses of the SWG waveguide with ${H_{\textrm{BOX}}} = $ 2 µm with the measured transmittance, as shown in Fig. 9(b). The estimated propagation loss of the fabricated Si-wire and SWG waveguides at the central wavelengths of the measured spectra are 1.1 dB/cm and 8.5 dB/cm, respectively.

 figure: Fig. 9.

Fig. 9. (a) Measured transmittances of the Si-wire filter (blue) and a Si-wire core waveguide without loading segments (black). (b) Measured transmittances of the SWG filter (red) and an SWG waveguide without loading segments (black). Simulated transmittance of the SWG waveguide in SOI with a 2 µm BOX (green). Losses of measurement setup, edge couplers and interconnecting waveguides have been subtracted from the experimental curves.

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In order to facilitate the comparison between the measured transmittance and the design target spectrum, in Fig. 10 we plot the normalized filter transmittance, i.e. corrected for the wavelength-dependent insertion loss of the waveguide core. The x-axis coordinate is defined as the relative wavelength, $\mathrm{\Delta }\lambda = \lambda - {\lambda _{11}}$, with ${\lambda _{11}}$ corresponding to the wavelength of the 11th notch of the spectral response (from shorter towards longer wavelength), i.e. ${\lambda _{11}}\; $ positioned approximately near the central wavelength. The Si-wire filter exhibits extinction ratio (ER) ranging from 9 to 16 dB. As expected, the relative peaks positions are less accurate for the outer peaks, with a maximum deviation of less than 200 pm. The measured 3-dB linewidths of the notches are in the range of 220 - 470 pm. The SWG filter exhibits reduced extinction ratios, from 7 to 11 dB, and the maximum wavelength peak deviation is comparable to the Si-wire filter. The 3-dB linewidths are within the range 210 - 340 pm. The discrepancy between simulation and experimental results is attributed to the reduction of phase coherence along the filter due to small deviations of the waveguide width and thickness [55]. These deviations are known to play a critical role in long silicon-based filters [56]. The performance of both filters might be improved by implementing them in a compact spiral geometry, thus reducing the effect of fabrication non-uniformities [57,58]. The worse extinction ratios exhibited by the SWG filter are attributed to its greater length, that makes this filter more sensitive to the loss of phase coherence, and its higher propagation loss due to substrate leakage, that could be mitigated by using a 3 µm BOX layer.

 figure: Fig. 10.

Fig. 10. Measured normalized transmittances of (a) the Si-wire filter (blue) and (b) the SWG filter (red). The corresponding target spectra are shown as black curves.

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6. Conclusions

We have proposed, investigated and experimentally demonstrated a new type of complex spectral filter in silicon waveguides. Our filter comprises a waveguide core with a delocalized mode and lateral loading segments physically separated from the core, forming a Bragg grating. The separation between the loading segments and the core and hence the grating coupling coefficient is modulated along the grating, to synthesize a complex spectral transmittance. The proposed strategy has been validated by designing and experimentally evaluating a proof-of-concept astrophotonic filter comprising 20 non-uniformly spaced notches matching OH emission lines near 1550 nm. To our knowledge this is the first experimental demonstration of such a complex spectral filter in silicon waveguides. We believe that our results open promising prospects for the development of nanophotonic waveguide filters with complex passband characteristics for spectral manipulation in integrated photonic devices on the SOI platform.

Funding

Ministerio de Economía y Competitividad (PID2019-106747RB-I00, TEC2016-80718-R); Ministerio de Educación, Cultura y Deporte (FPU16/03401, FPU16/06762, FPU17/00638); Junta de Andalucía (P18-RT-1453, P18-RT-793, UMA18-FEDERJA-219); Universidad de Málaga; National Research Council Canada (CSTIP grant #HTSN210); Grantová Agentura České Republiky (19-00062S).

Disclosures

The authors declare no conflicts of interest.

References

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2. A. Othonos, “Fiber Bragg Gratings,” Rev. Sci. Instrum. 68(12), 4309–4341 (1997). [CrossRef]  

3. R. Kashyap, Fiber Bragg Gratings (Academic press, 2009).

4. X. Wang, W. Shi, R. Vafaei, N. A. F. Jaeger, and L. Chrostowski, “Uniform and Sampled Bragg Gratings in SOI Strip Waveguides with Sidewall Corrugations,” IEEE Photonics Technol. Lett. 23(5), 290–292 (2011). [CrossRef]  

5. A. D. Simard, N. Belhadj, Y. Painchaud, and S. LaRochelle, “Apodized Silicon-on-Insulator Bragg Gratings,” IEEE Photonics Technol. Lett. 24(12), 1033–1035 (2012). [CrossRef]  

6. I. W. Frank, Y. Zhang, and M. Loncar, “Nearly arbitrary on-chip optical filters for ultrafast pulse shaping,” Opt. Express 22(19), 22403–22410 (2014). [CrossRef]  

7. M. J. Strain, S. Thoms, D. S. MacIntyre, and M. Sorel, “Multi-wavelength filters in silicon using superposition sidewall Bragg grating devices,” Opt. Lett. 39(2), 413–416 (2014). [CrossRef]  

8. M. Verbist, W. Bogaerts, and D. Van Thourhout, “Design of Weak 1-D Bragg Grating Filters in SOI Waveguides Using Volume Holography Techniques,” J. Lightwave Technol. 32(10), 1915–1920 (2014). [CrossRef]  

9. X. Wang, Y. Wang, J. Flueckiger, R. Bojko, A. Liu, A. Reid, J. Pond, N. A. F. Jaeger, and L. Chrostowski, “Precise control of the coupling coefficient through destructive interference in silicon waveguide Bragg gratings,” Opt. Lett. 39(19), 5519–5522 (2014). [CrossRef]  

10. M. Caverley, X. Wang, K. Murray, N. A. F. Jaeger, and L. Chrostowski, “Silicon-on-Insulator Modulators Using a Quarter-Wave Phase-Shifted Bragg Grating,” IEEE Photonics Technol. Lett. 27(22), 2331–2334 (2015). [CrossRef]  

11. J. Wang and L. R. Chen, “Low crosstalk Bragg grating/Mach-Zehnder interferometer optical add-drop multiplexer in silicon photonics,” Opt. Express 23(20), 26450–26459 (2015). [CrossRef]  

12. A. D. Simard, M. J. Strain, L. Meriggi, M. Sorel, and S. LaRochelle, “Bandpass integrated Bragg gratings in silicon-on-insulator with well-controlled amplitude and phase responses,” Opt. Lett. 40(5), 736–739 (2015). [CrossRef]  

13. H. P. Bazargani, M. Burla, L. Chrostowski, and J. Azaña, “Photonic Hilbert transformer based on laterally apodized integrated waveguide Bragg gratings on a SOI wafer,” Opt. Lett. 41(21), 5039–5042 (2016). [CrossRef]  

14. S. Paul, T. Saastamoinen, S. Honkanen, M. Roussey, and M. Kuittinen, “Multi-wavelength filtering with a waveguide integrated phase-modulated Bragg grating,” Opt. Lett. 42(22), 4635–4638 (2017). [CrossRef]  

15. R. Cheng and L. Chrostowski, “Multichannel photonic Hilbert transformers based on complex modulated integrated Bragg gratings,” Opt. Lett. 43(5), 1031–1034 (2018). [CrossRef]  

16. C. Feng, R. Soref, R. T. Chen, X. Xu, and W. Jiang, “Efficient and accurate synthesis of complex Bragg grating waveguide in dispersive silicon structures,” J. Opt. Soc. Am. B 35(8), 1921–1927 (2018). [CrossRef]  

17. S. Kaushal, R. Cheng, M. Ma, A. Mistry, M. Burla, L. Chrostowski, and J. Azaña, “Optical signal processing based on silicon photonics waveguide Bragg gratings: review,” Front. Optoelectron. 11(2), 163–188 (2018). [CrossRef]  

18. R. Cheng, H. Yun, S. Lin, Y. Han, and L. Chrostowski, “Apodization profile amplification of silicon integrated Bragg gratings through lateral phase delays,” Opt. Lett. 44(2), 435–438 (2019). [CrossRef]  

19. R. Cheng and L. Chrostowski, “Apodization of Silicon Integrated Bragg Gratings Through Periodic Phase Modulation,” IEEE J. Sel. Top. Quantum Electron. 25(5), 1–11 (2019). [CrossRef]  

20. R. Cheng, N. A. F. Jaeger, and L. Chrostowski, “Fully tailorable integrated-optic resonators based on chirped waveguide Moiré gratings,” Optica 7(6), 647–657 (2020). [CrossRef]  

21. D. Pérez-Galacho, C. Alonso-Ramos, F. Mazeas, X. Le Roux, D. Oser, W. Zhang, D. Marris-Morini, L. Labonté, S. Tanzilli, É. Cassan, and L. Vivien, “Optical pump-rejection filter based on silicon sub-wavelength engineered photonic structures,” Opt. Lett. 42(8), 1468–1471 (2017). [CrossRef]  

22. D. Oser, D. Pérez-Galacho, C. Alonso-Ramos, X. Le Roux, S. Tanzilli, L. Vivien, L. Labonté, and É. Cassan, “Subwavelength engineering and asymmetry: two efficient tools for sub-nanometer-bandwidth silicon Bragg filters,” Opt. Lett. 43(14), 3208–3211 (2018). [CrossRef]  

23. D. Oser, D. Pérez-Galacho, X. Le Roux, S. Tanzilli, L. Vivien, L. Labonté, É. Cassan, and C. Alonso-Ramos, “Silicon subwavelength modal Bragg grating filters with narrow bandwidth and high optical rejection,” Opt. Lett. 45(20), 5784–5787 (2020). [CrossRef]  

24. D. T. H. Tan, K. Ikeda, and Y. Fainman, “Cladding-modulated Bragg gratings in silicon waveguides,” Opt. Lett. 34(9), 1357–1359 (2009). [CrossRef]  

25. S. Zamek, D. T. H. Tan, M. Khajavikhan, M. Ayache, M. P. Nezhad, and Y. Fainman, “Compact chip-scale filter based on curved waveguide Bragg gratings,” Opt. Lett. 35(20), 3477–3479 (2010). [CrossRef]  

26. Y.-J. Hung, K.-H. Lin, C.-J. Wu, C.-Y. Wang, and Y.-J. Chen, “Narrowband Reflection From Weakly Coupled Cladding-Modulated Bragg Gratings,” IEEE J. Sel. Top. Quantum Electron. 22(6), 218–224 (2016). [CrossRef]  

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29. J. Čtyroký, J. G. Wangüemert-Pérez, P. Kwiecien, I. Richter, J. Litvik, J. H. Schmid, Í Molina-Fernández, A. Ortega-Moñux, M. Dado, and P. Cheben, “Design of narrowband Bragg spectral filters in subwavelength grating metamaterial waveguides,” Opt. Express 26(1), 179–194 (2018). [CrossRef]  

30. P. Cheben, J. Čtyroký, J. H. Schmid, S. Wang, J. Lapointe, J. G. Wangüemert-Pérez, Í Molina-Fernández, A. Ortega-Moñux, R. Halir, D. Melati, D. Xu, S. Janz, and M. Dado, “Bragg filter bandwidth engineering in subwavelength grating metamaterial waveguides,” Opt. Lett. 44(4), 1043–1046 (2019). [CrossRef]  

31. H. Yun, M. Hammood, L. Chrostowski, and N. A. F. Jaeger, “Optical Add-drop Filters using Cladding-modulated Sub-wavelength Grating Contra-directional Couplers for Silicon-on-Insulator Platforms,” in IEEE 10th Annual Information Technology, Electronics and Mobile Communication Conference (IEMCON) (2019), pp. 0926–0931.

32. H. Sun, Y. Wang, and L. R. Chen, “Integrated Discretely Tunable Optical Delay Line Based on Step-Chirped Subwavelength Grating Waveguide Bragg Gratings,” J. Lightwave Technol. 38(19), 5551–5560 (2020). [CrossRef]  

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34. S. Xie, J. Zhan, Y. Hu, Y. Zhang, S. Veilleux, J. Bland-Hawthorn, and M. Dagenais, “Add-drop filter with complex waveguide Bragg grating and multimode interferometer operating on arbitrarily spaced channels,” Opt. Lett. 43(24), 6045–6048 (2018). [CrossRef]  

35. Y. W. Hu, S. Xie, J. Zhan, Y. Zhang, S. Veilleux, and M. Dagenais, “Integrated Arbitrary Filter with Spiral Gratings: Design and Characterization,” J. Lightwave Technol. 38(16), 4454–4461 (2020). [CrossRef]  

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37. J. Skaar, L. Wang, and T. Erdogan, “On the Synthesis of Fiber Bragg Gratings by Layer Peeling,” IEEE J. Quantum Electron. 37(2), 165–173 (2001). [CrossRef]  

38. P. Cheben, D.-X. Xu, S. Janz, and A. Densmore, “Subwavelength waveguide grating for mode conversion and light coupling in integrated optics,” Opt. Express 14(11), 4695–4702 (2006). [CrossRef]  

39. J. H. Schmid, P. Cheben, S. Janz, J. Lapointe, E. Post, and D.-X. Xu, “Gradient-index antireflective subwavelength structures for planar waveguide facets,” Opt. Lett. 32(13), 1794–1796 (2007). [CrossRef]  

40. J. H. Schmid, P. Cheben, S. Janz, J. Lapointe, E. Post, A. Delâge, A. Densmore, B. Lamontagne, P. Waldron, and D. X. Xu, “Subwavelength Grating Structures in Silicon-on-Insulator waveguides,” Adv. Opt. Technol. 2008 (2008).

41. P. J. Bock, P. Cheben, J. H. Schmid, A. Delâge, D.-X. Xu, S. Janz, and T. J. Hall, “Sub-wavelength grating mode transformers in silicon slab waveguides,” Opt. Express 17(21), 19120–19133 (2009). [CrossRef]  

42. J. H. Schmid, P. Cheben, P. J. Bock, R. Halir, J. Lapointe, S. Janz, A. Delâge, A. Densmore, J.-M. Fédeli, T. J. Hall, B. Lamontagne, R. Ma, I. Molina-Fernández, and D.-X. Xu, “Refractive Index Engineering With Subwavelength Gratings in Silicon Microphotonic Waveguides,” IEEE Photonics J. 3(3), 597–607 (2011). [CrossRef]  

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44. R. Halir, P. J. Bock, P. Cheben, A. Ortega-Moñux, C. Alonso-Ramos, J. H. Schmid, J. Lapointe, D.-X. Xu, J. G. Wangüemert-Pérez, Í Molina-Fernández, and S. Janz, “Waveguide sub-wavelength structures: a review of principles and applications,” Laser Photonics Rev. 9(1), 25–49 (2015). [CrossRef]  

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46. R. Halir, A. Ortega-Moñux, D. Benedikovic, G. Z. Mashanovich, J. G. Wangüemert-Pérez, J. H. Schmid, Í Molina-Fernández, and P. Cheben, “Subwavelength-Grating Metamaterial Structures for Silicon Photonic Devices,” Proc. IEEE 106(12), 2144–2157 (2018). [CrossRef]  

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52. P. Cheben, S. Janz, D.-X. Xu, J. H. Schmid, A. Densmore, and J. Lapointe, “Interface device for performing mode transformation in optical waveguides,” U.S. patent 7,680,371 (16 March 2010).

53. P. Cheben, P. J. Bock, J. H. Schmid, J. Lapointe, S. Janz, D.-X. Xu, A. Densmore, A. Delâge, B. Lamontagne, and T. J. Hall, “Refractive index engineering with subwavelength gratings for efficient microphotonic couplers and planar waveguide multiplexers,” Opt. Lett. 35(15), 2526–2528 (2010). [CrossRef]  

54. P. Cheben, J. H. Schmid, S. Wang, D.-X. Xu, M. Vachon, S. Janz, J. Lapointe, Y. Painchaud, and M.-J. Picard, “Broadband polarization independent nanophotonic coupler for silicon waveguides with ultra-high efficiency,” Opt. Express 23(17), 22553–22563 (2015). [CrossRef]  

55. Z. Lu, J. Jhoja, J. Klein, X. Wang, A. Liu, J. Flueckiger, J. Pond, and L. Chrostowski, “Performance prediction for silicon photonics integrated circuits with layout-dependent correlated manufacturing variability,” Opt. Express 25(9), 9712–9733 (2017). [CrossRef]  

56. D. Oser, F. Mazeas, X. Le Roux, D. Pérez-Galacho, O. Alibart, S. Tanzilli, L. Labonté, D. Marris-Morini, L. Vivien, É. Cassan, and C. Alonso-Ramos, “Coherency-Broken Bragg Filters: Overcoming On-Chip Rejection Limitations,” Laser Photonics Rev. 13(8), 1800226 (2019). [CrossRef]  

57. A. D. Simard, Y. Painchaud, and S. LaRochelle, “Integrated Bragg gratings in spiral waveguides,” Opt. Express 21(7), 8953–8963 (2013). [CrossRef]  

58. Z. Chen, J. Flueckiger, X. Wang, F. Zhang, H. Yun, Z. Lu, M. Caverley, Y. Wang, N. A. F. Jaeger, and L. Chrostowski, “Spiral Bragg grating waveguides for TM mode silicon photonics,” Opt. Express 23(19), 25295–25307 (2015). [CrossRef]  

References

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  1. K. O. Hill and G. Meltz, “Fiber Bragg Grating Technology Fundamentals and Overview,” J. Lightwave Technol. 15(8), 1263–1276 (1997).
    [Crossref]
  2. A. Othonos, “Fiber Bragg Gratings,” Rev. Sci. Instrum. 68(12), 4309–4341 (1997).
    [Crossref]
  3. R. Kashyap, Fiber Bragg Gratings (Academic press, 2009).
  4. X. Wang, W. Shi, R. Vafaei, N. A. F. Jaeger, and L. Chrostowski, “Uniform and Sampled Bragg Gratings in SOI Strip Waveguides with Sidewall Corrugations,” IEEE Photonics Technol. Lett. 23(5), 290–292 (2011).
    [Crossref]
  5. A. D. Simard, N. Belhadj, Y. Painchaud, and S. LaRochelle, “Apodized Silicon-on-Insulator Bragg Gratings,” IEEE Photonics Technol. Lett. 24(12), 1033–1035 (2012).
    [Crossref]
  6. I. W. Frank, Y. Zhang, and M. Loncar, “Nearly arbitrary on-chip optical filters for ultrafast pulse shaping,” Opt. Express 22(19), 22403–22410 (2014).
    [Crossref]
  7. M. J. Strain, S. Thoms, D. S. MacIntyre, and M. Sorel, “Multi-wavelength filters in silicon using superposition sidewall Bragg grating devices,” Opt. Lett. 39(2), 413–416 (2014).
    [Crossref]
  8. M. Verbist, W. Bogaerts, and D. Van Thourhout, “Design of Weak 1-D Bragg Grating Filters in SOI Waveguides Using Volume Holography Techniques,” J. Lightwave Technol. 32(10), 1915–1920 (2014).
    [Crossref]
  9. X. Wang, Y. Wang, J. Flueckiger, R. Bojko, A. Liu, A. Reid, J. Pond, N. A. F. Jaeger, and L. Chrostowski, “Precise control of the coupling coefficient through destructive interference in silicon waveguide Bragg gratings,” Opt. Lett. 39(19), 5519–5522 (2014).
    [Crossref]
  10. M. Caverley, X. Wang, K. Murray, N. A. F. Jaeger, and L. Chrostowski, “Silicon-on-Insulator Modulators Using a Quarter-Wave Phase-Shifted Bragg Grating,” IEEE Photonics Technol. Lett. 27(22), 2331–2334 (2015).
    [Crossref]
  11. J. Wang and L. R. Chen, “Low crosstalk Bragg grating/Mach-Zehnder interferometer optical add-drop multiplexer in silicon photonics,” Opt. Express 23(20), 26450–26459 (2015).
    [Crossref]
  12. A. D. Simard, M. J. Strain, L. Meriggi, M. Sorel, and S. LaRochelle, “Bandpass integrated Bragg gratings in silicon-on-insulator with well-controlled amplitude and phase responses,” Opt. Lett. 40(5), 736–739 (2015).
    [Crossref]
  13. H. P. Bazargani, M. Burla, L. Chrostowski, and J. Azaña, “Photonic Hilbert transformer based on laterally apodized integrated waveguide Bragg gratings on a SOI wafer,” Opt. Lett. 41(21), 5039–5042 (2016).
    [Crossref]
  14. S. Paul, T. Saastamoinen, S. Honkanen, M. Roussey, and M. Kuittinen, “Multi-wavelength filtering with a waveguide integrated phase-modulated Bragg grating,” Opt. Lett. 42(22), 4635–4638 (2017).
    [Crossref]
  15. R. Cheng and L. Chrostowski, “Multichannel photonic Hilbert transformers based on complex modulated integrated Bragg gratings,” Opt. Lett. 43(5), 1031–1034 (2018).
    [Crossref]
  16. C. Feng, R. Soref, R. T. Chen, X. Xu, and W. Jiang, “Efficient and accurate synthesis of complex Bragg grating waveguide in dispersive silicon structures,” J. Opt. Soc. Am. B 35(8), 1921–1927 (2018).
    [Crossref]
  17. S. Kaushal, R. Cheng, M. Ma, A. Mistry, M. Burla, L. Chrostowski, and J. Azaña, “Optical signal processing based on silicon photonics waveguide Bragg gratings: review,” Front. Optoelectron. 11(2), 163–188 (2018).
    [Crossref]
  18. R. Cheng, H. Yun, S. Lin, Y. Han, and L. Chrostowski, “Apodization profile amplification of silicon integrated Bragg gratings through lateral phase delays,” Opt. Lett. 44(2), 435–438 (2019).
    [Crossref]
  19. R. Cheng and L. Chrostowski, “Apodization of Silicon Integrated Bragg Gratings Through Periodic Phase Modulation,” IEEE J. Sel. Top. Quantum Electron. 25(5), 1–11 (2019).
    [Crossref]
  20. R. Cheng, N. A. F. Jaeger, and L. Chrostowski, “Fully tailorable integrated-optic resonators based on chirped waveguide Moiré gratings,” Optica 7(6), 647–657 (2020).
    [Crossref]
  21. D. Pérez-Galacho, C. Alonso-Ramos, F. Mazeas, X. Le Roux, D. Oser, W. Zhang, D. Marris-Morini, L. Labonté, S. Tanzilli, É. Cassan, and L. Vivien, “Optical pump-rejection filter based on silicon sub-wavelength engineered photonic structures,” Opt. Lett. 42(8), 1468–1471 (2017).
    [Crossref]
  22. D. Oser, D. Pérez-Galacho, C. Alonso-Ramos, X. Le Roux, S. Tanzilli, L. Vivien, L. Labonté, and É. Cassan, “Subwavelength engineering and asymmetry: two efficient tools for sub-nanometer-bandwidth silicon Bragg filters,” Opt. Lett. 43(14), 3208–3211 (2018).
    [Crossref]
  23. D. Oser, D. Pérez-Galacho, X. Le Roux, S. Tanzilli, L. Vivien, L. Labonté, É. Cassan, and C. Alonso-Ramos, “Silicon subwavelength modal Bragg grating filters with narrow bandwidth and high optical rejection,” Opt. Lett. 45(20), 5784–5787 (2020).
    [Crossref]
  24. D. T. H. Tan, K. Ikeda, and Y. Fainman, “Cladding-modulated Bragg gratings in silicon waveguides,” Opt. Lett. 34(9), 1357–1359 (2009).
    [Crossref]
  25. S. Zamek, D. T. H. Tan, M. Khajavikhan, M. Ayache, M. P. Nezhad, and Y. Fainman, “Compact chip-scale filter based on curved waveguide Bragg gratings,” Opt. Lett. 35(20), 3477–3479 (2010).
    [Crossref]
  26. Y.-J. Hung, K.-H. Lin, C.-J. Wu, C.-Y. Wang, and Y.-J. Chen, “Narrowband Reflection From Weakly Coupled Cladding-Modulated Bragg Gratings,” IEEE J. Sel. Top. Quantum Electron. 22(6), 218–224 (2016).
    [Crossref]
  27. H. Qiu, L. Lin, P. Yu, T. Dai, X. Jiang, and H. Yu, “Narrow-Band Add-Drop Filter Based on Cladding-Modulated Apodized Multimode Bragg Grating,” J. Lightwave Technol. 37(21), 5542–5547 (2019).
    [Crossref]
  28. T.-H. Yen and Y.-J. Hung, “Narrowband Dual-Wavelength Silicon Waveguide Bragg Reflectors,” J. Lightwave Technol. 37(20), 5326–5332 (2019).
    [Crossref]
  29. J. Čtyroký, J. G. Wangüemert-Pérez, P. Kwiecien, I. Richter, J. Litvik, J. H. Schmid, Í Molina-Fernández, A. Ortega-Moñux, M. Dado, and P. Cheben, “Design of narrowband Bragg spectral filters in subwavelength grating metamaterial waveguides,” Opt. Express 26(1), 179–194 (2018).
    [Crossref]
  30. P. Cheben, J. Čtyroký, J. H. Schmid, S. Wang, J. Lapointe, J. G. Wangüemert-Pérez, Í Molina-Fernández, A. Ortega-Moñux, R. Halir, D. Melati, D. Xu, S. Janz, and M. Dado, “Bragg filter bandwidth engineering in subwavelength grating metamaterial waveguides,” Opt. Lett. 44(4), 1043–1046 (2019).
    [Crossref]
  31. H. Yun, M. Hammood, L. Chrostowski, and N. A. F. Jaeger, “Optical Add-drop Filters using Cladding-modulated Sub-wavelength Grating Contra-directional Couplers for Silicon-on-Insulator Platforms,” in IEEE 10th Annual Information Technology, Electronics and Mobile Communication Conference (IEMCON) (2019), pp. 0926–0931.
  32. H. Sun, Y. Wang, and L. R. Chen, “Integrated Discretely Tunable Optical Delay Line Based on Step-Chirped Subwavelength Grating Waveguide Bragg Gratings,” J. Lightwave Technol. 38(19), 5551–5560 (2020).
    [Crossref]
  33. T. Zhu, Y. Hu, P. Gatkine, S. Veilleux, J. Bland-Hawthorn, and M. Dagenais, “Arbitrary on-chip optical filter using complex waveguide Bragg gratings,” Appl. Phys. Lett. 108(10), 101104 (2016).
    [Crossref]
  34. S. Xie, J. Zhan, Y. Hu, Y. Zhang, S. Veilleux, J. Bland-Hawthorn, and M. Dagenais, “Add-drop filter with complex waveguide Bragg grating and multimode interferometer operating on arbitrarily spaced channels,” Opt. Lett. 43(24), 6045–6048 (2018).
    [Crossref]
  35. Y. W. Hu, S. Xie, J. Zhan, Y. Zhang, S. Veilleux, and M. Dagenais, “Integrated Arbitrary Filter with Spiral Gratings: Design and Characterization,” J. Lightwave Technol. 38(16), 4454–4461 (2020).
    [Crossref]
  36. R. Feced, M. N. Zervas, and M. A. Muriel, “Efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings,” IEEE J. Quantum Electron. 35(8), 1105–1115 (1999).
    [Crossref]
  37. J. Skaar, L. Wang, and T. Erdogan, “On the Synthesis of Fiber Bragg Gratings by Layer Peeling,” IEEE J. Quantum Electron. 37(2), 165–173 (2001).
    [Crossref]
  38. P. Cheben, D.-X. Xu, S. Janz, and A. Densmore, “Subwavelength waveguide grating for mode conversion and light coupling in integrated optics,” Opt. Express 14(11), 4695–4702 (2006).
    [Crossref]
  39. J. H. Schmid, P. Cheben, S. Janz, J. Lapointe, E. Post, and D.-X. Xu, “Gradient-index antireflective subwavelength structures for planar waveguide facets,” Opt. Lett. 32(13), 1794–1796 (2007).
    [Crossref]
  40. J. H. Schmid, P. Cheben, S. Janz, J. Lapointe, E. Post, A. Delâge, A. Densmore, B. Lamontagne, P. Waldron, and D. X. Xu, “Subwavelength Grating Structures in Silicon-on-Insulator waveguides,” Adv. Opt. Technol. 2008 (2008).
  41. P. J. Bock, P. Cheben, J. H. Schmid, A. Delâge, D.-X. Xu, S. Janz, and T. J. Hall, “Sub-wavelength grating mode transformers in silicon slab waveguides,” Opt. Express 17(21), 19120–19133 (2009).
    [Crossref]
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  53. P. Cheben, P. J. Bock, J. H. Schmid, J. Lapointe, S. Janz, D.-X. Xu, A. Densmore, A. Delâge, B. Lamontagne, and T. J. Hall, “Refractive index engineering with subwavelength gratings for efficient microphotonic couplers and planar waveguide multiplexers,” Opt. Lett. 35(15), 2526–2528 (2010).
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2020 (4)

2019 (6)

2018 (8)

R. Halir, A. Ortega-Moñux, D. Benedikovic, G. Z. Mashanovich, J. G. Wangüemert-Pérez, J. H. Schmid, Í Molina-Fernández, and P. Cheben, “Subwavelength-Grating Metamaterial Structures for Silicon Photonic Devices,” Proc. IEEE 106(12), 2144–2157 (2018).
[Crossref]

P. Cheben, R. Halir, J. H. Schmid, H. A. Atwater, and D. R. Smith, “Subwavelength integrated photonics,” Nature 560(7720), 565–572 (2018).
[Crossref]

R. Cheng and L. Chrostowski, “Multichannel photonic Hilbert transformers based on complex modulated integrated Bragg gratings,” Opt. Lett. 43(5), 1031–1034 (2018).
[Crossref]

C. Feng, R. Soref, R. T. Chen, X. Xu, and W. Jiang, “Efficient and accurate synthesis of complex Bragg grating waveguide in dispersive silicon structures,” J. Opt. Soc. Am. B 35(8), 1921–1927 (2018).
[Crossref]

S. Kaushal, R. Cheng, M. Ma, A. Mistry, M. Burla, L. Chrostowski, and J. Azaña, “Optical signal processing based on silicon photonics waveguide Bragg gratings: review,” Front. Optoelectron. 11(2), 163–188 (2018).
[Crossref]

J. Čtyroký, J. G. Wangüemert-Pérez, P. Kwiecien, I. Richter, J. Litvik, J. H. Schmid, Í Molina-Fernández, A. Ortega-Moñux, M. Dado, and P. Cheben, “Design of narrowband Bragg spectral filters in subwavelength grating metamaterial waveguides,” Opt. Express 26(1), 179–194 (2018).
[Crossref]

D. Oser, D. Pérez-Galacho, C. Alonso-Ramos, X. Le Roux, S. Tanzilli, L. Vivien, L. Labonté, and É. Cassan, “Subwavelength engineering and asymmetry: two efficient tools for sub-nanometer-bandwidth silicon Bragg filters,” Opt. Lett. 43(14), 3208–3211 (2018).
[Crossref]

S. Xie, J. Zhan, Y. Hu, Y. Zhang, S. Veilleux, J. Bland-Hawthorn, and M. Dagenais, “Add-drop filter with complex waveguide Bragg grating and multimode interferometer operating on arbitrarily spaced channels,” Opt. Lett. 43(24), 6045–6048 (2018).
[Crossref]

2017 (4)

2016 (4)

J. D. Sarmiento-Merenguel, A. Ortega-Moñux, J.-M. Fédéli, J. G. Wangüemert-Pérez, C. Alonso-Ramos, E. Durán-Valdeiglesias, P. Cheben, Í Molina-Fernández, and R. Halir, “Controlling leakage losses in subwavelength grating silicon metamaterial waveguides,” Opt. Lett. 41(15), 3443–3446 (2016).
[Crossref]

H. P. Bazargani, M. Burla, L. Chrostowski, and J. Azaña, “Photonic Hilbert transformer based on laterally apodized integrated waveguide Bragg gratings on a SOI wafer,” Opt. Lett. 41(21), 5039–5042 (2016).
[Crossref]

Y.-J. Hung, K.-H. Lin, C.-J. Wu, C.-Y. Wang, and Y.-J. Chen, “Narrowband Reflection From Weakly Coupled Cladding-Modulated Bragg Gratings,” IEEE J. Sel. Top. Quantum Electron. 22(6), 218–224 (2016).
[Crossref]

T. Zhu, Y. Hu, P. Gatkine, S. Veilleux, J. Bland-Hawthorn, and M. Dagenais, “Arbitrary on-chip optical filter using complex waveguide Bragg gratings,” Appl. Phys. Lett. 108(10), 101104 (2016).
[Crossref]

2015 (6)

2014 (4)

2013 (1)

2012 (2)

J. Čtyroký, “3-D Bidirectional Propagation Algorithm Based on Fourier Series,” J. Lightwave Technol. 30(23), 3699–3708 (2012).
[Crossref]

A. D. Simard, N. Belhadj, Y. Painchaud, and S. LaRochelle, “Apodized Silicon-on-Insulator Bragg Gratings,” IEEE Photonics Technol. Lett. 24(12), 1033–1035 (2012).
[Crossref]

2011 (2)

X. Wang, W. Shi, R. Vafaei, N. A. F. Jaeger, and L. Chrostowski, “Uniform and Sampled Bragg Gratings in SOI Strip Waveguides with Sidewall Corrugations,” IEEE Photonics Technol. Lett. 23(5), 290–292 (2011).
[Crossref]

J. H. Schmid, P. Cheben, P. J. Bock, R. Halir, J. Lapointe, S. Janz, A. Delâge, A. Densmore, J.-M. Fédeli, T. J. Hall, B. Lamontagne, R. Ma, I. Molina-Fernández, and D.-X. Xu, “Refractive Index Engineering With Subwavelength Gratings in Silicon Microphotonic Waveguides,” IEEE Photonics J. 3(3), 597–607 (2011).
[Crossref]

2010 (2)

2009 (2)

2007 (1)

2006 (1)

2001 (1)

J. Skaar, L. Wang, and T. Erdogan, “On the Synthesis of Fiber Bragg Gratings by Layer Peeling,” IEEE J. Quantum Electron. 37(2), 165–173 (2001).
[Crossref]

1999 (1)

R. Feced, M. N. Zervas, and M. A. Muriel, “Efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings,” IEEE J. Quantum Electron. 35(8), 1105–1115 (1999).
[Crossref]

1997 (2)

K. O. Hill and G. Meltz, “Fiber Bragg Grating Technology Fundamentals and Overview,” J. Lightwave Technol. 15(8), 1263–1276 (1997).
[Crossref]

A. Othonos, “Fiber Bragg Gratings,” Rev. Sci. Instrum. 68(12), 4309–4341 (1997).
[Crossref]

Alibart, O.

D. Oser, F. Mazeas, X. Le Roux, D. Pérez-Galacho, O. Alibart, S. Tanzilli, L. Labonté, D. Marris-Morini, L. Vivien, É. Cassan, and C. Alonso-Ramos, “Coherency-Broken Bragg Filters: Overcoming On-Chip Rejection Limitations,” Laser Photonics Rev. 13(8), 1800226 (2019).
[Crossref]

Alonso-Ramos, C.

D. Oser, D. Pérez-Galacho, X. Le Roux, S. Tanzilli, L. Vivien, L. Labonté, É. Cassan, and C. Alonso-Ramos, “Silicon subwavelength modal Bragg grating filters with narrow bandwidth and high optical rejection,” Opt. Lett. 45(20), 5784–5787 (2020).
[Crossref]

D. Oser, F. Mazeas, X. Le Roux, D. Pérez-Galacho, O. Alibart, S. Tanzilli, L. Labonté, D. Marris-Morini, L. Vivien, É. Cassan, and C. Alonso-Ramos, “Coherency-Broken Bragg Filters: Overcoming On-Chip Rejection Limitations,” Laser Photonics Rev. 13(8), 1800226 (2019).
[Crossref]

D. Oser, D. Pérez-Galacho, C. Alonso-Ramos, X. Le Roux, S. Tanzilli, L. Vivien, L. Labonté, and É. Cassan, “Subwavelength engineering and asymmetry: two efficient tools for sub-nanometer-bandwidth silicon Bragg filters,” Opt. Lett. 43(14), 3208–3211 (2018).
[Crossref]

D. Pérez-Galacho, C. Alonso-Ramos, F. Mazeas, X. Le Roux, D. Oser, W. Zhang, D. Marris-Morini, L. Labonté, S. Tanzilli, É. Cassan, and L. Vivien, “Optical pump-rejection filter based on silicon sub-wavelength engineered photonic structures,” Opt. Lett. 42(8), 1468–1471 (2017).
[Crossref]

J. D. Sarmiento-Merenguel, A. Ortega-Moñux, J.-M. Fédéli, J. G. Wangüemert-Pérez, C. Alonso-Ramos, E. Durán-Valdeiglesias, P. Cheben, Í Molina-Fernández, and R. Halir, “Controlling leakage losses in subwavelength grating silicon metamaterial waveguides,” Opt. Lett. 41(15), 3443–3446 (2016).
[Crossref]

R. Halir, P. J. Bock, P. Cheben, A. Ortega-Moñux, C. Alonso-Ramos, J. H. Schmid, J. Lapointe, D.-X. Xu, J. G. Wangüemert-Pérez, Í Molina-Fernández, and S. Janz, “Waveguide sub-wavelength structures: a review of principles and applications,” Laser Photonics Rev. 9(1), 25–49 (2015).
[Crossref]

Atwater, H. A.

P. Cheben, R. Halir, J. H. Schmid, H. A. Atwater, and D. R. Smith, “Subwavelength integrated photonics,” Nature 560(7720), 565–572 (2018).
[Crossref]

Ayache, M.

Azaña, J.

S. Kaushal, R. Cheng, M. Ma, A. Mistry, M. Burla, L. Chrostowski, and J. Azaña, “Optical signal processing based on silicon photonics waveguide Bragg gratings: review,” Front. Optoelectron. 11(2), 163–188 (2018).
[Crossref]

H. P. Bazargani, M. Burla, L. Chrostowski, and J. Azaña, “Photonic Hilbert transformer based on laterally apodized integrated waveguide Bragg gratings on a SOI wafer,” Opt. Lett. 41(21), 5039–5042 (2016).
[Crossref]

Bazargani, H. P.

Belhadj, N.

A. D. Simard, N. Belhadj, Y. Painchaud, and S. LaRochelle, “Apodized Silicon-on-Insulator Bragg Gratings,” IEEE Photonics Technol. Lett. 24(12), 1033–1035 (2012).
[Crossref]

Benedikovic, D.

R. Halir, A. Ortega-Moñux, D. Benedikovic, G. Z. Mashanovich, J. G. Wangüemert-Pérez, J. H. Schmid, Í Molina-Fernández, and P. Cheben, “Subwavelength-Grating Metamaterial Structures for Silicon Photonic Devices,” Proc. IEEE 106(12), 2144–2157 (2018).
[Crossref]

Bland-Hawthorn, J.

S. Xie, J. Zhan, Y. Hu, Y. Zhang, S. Veilleux, J. Bland-Hawthorn, and M. Dagenais, “Add-drop filter with complex waveguide Bragg grating and multimode interferometer operating on arbitrarily spaced channels,” Opt. Lett. 43(24), 6045–6048 (2018).
[Crossref]

T. Zhu, Y. Hu, P. Gatkine, S. Veilleux, J. Bland-Hawthorn, and M. Dagenais, “Arbitrary on-chip optical filter using complex waveguide Bragg gratings,” Appl. Phys. Lett. 108(10), 101104 (2016).
[Crossref]

Bock, P. J.

R. Halir, P. J. Bock, P. Cheben, A. Ortega-Moñux, C. Alonso-Ramos, J. H. Schmid, J. Lapointe, D.-X. Xu, J. G. Wangüemert-Pérez, Í Molina-Fernández, and S. Janz, “Waveguide sub-wavelength structures: a review of principles and applications,” Laser Photonics Rev. 9(1), 25–49 (2015).
[Crossref]

J. H. Schmid, P. Cheben, P. J. Bock, R. Halir, J. Lapointe, S. Janz, A. Delâge, A. Densmore, J.-M. Fédeli, T. J. Hall, B. Lamontagne, R. Ma, I. Molina-Fernández, and D.-X. Xu, “Refractive Index Engineering With Subwavelength Gratings in Silicon Microphotonic Waveguides,” IEEE Photonics J. 3(3), 597–607 (2011).
[Crossref]

P. Cheben, P. J. Bock, J. H. Schmid, J. Lapointe, S. Janz, D.-X. Xu, A. Densmore, A. Delâge, B. Lamontagne, and T. J. Hall, “Refractive index engineering with subwavelength gratings for efficient microphotonic couplers and planar waveguide multiplexers,” Opt. Lett. 35(15), 2526–2528 (2010).
[Crossref]

P. J. Bock, P. Cheben, J. H. Schmid, A. Delâge, D.-X. Xu, S. Janz, and T. J. Hall, “Sub-wavelength grating mode transformers in silicon slab waveguides,” Opt. Express 17(21), 19120–19133 (2009).
[Crossref]

Bogaerts, W.

Bojko, R.

Burla, M.

S. Kaushal, R. Cheng, M. Ma, A. Mistry, M. Burla, L. Chrostowski, and J. Azaña, “Optical signal processing based on silicon photonics waveguide Bragg gratings: review,” Front. Optoelectron. 11(2), 163–188 (2018).
[Crossref]

H. P. Bazargani, M. Burla, L. Chrostowski, and J. Azaña, “Photonic Hilbert transformer based on laterally apodized integrated waveguide Bragg gratings on a SOI wafer,” Opt. Lett. 41(21), 5039–5042 (2016).
[Crossref]

Cassan, É.

Caverley, M.

M. Caverley, X. Wang, K. Murray, N. A. F. Jaeger, and L. Chrostowski, “Silicon-on-Insulator Modulators Using a Quarter-Wave Phase-Shifted Bragg Grating,” IEEE Photonics Technol. Lett. 27(22), 2331–2334 (2015).
[Crossref]

Z. Chen, J. Flueckiger, X. Wang, F. Zhang, H. Yun, Z. Lu, M. Caverley, Y. Wang, N. A. F. Jaeger, and L. Chrostowski, “Spiral Bragg grating waveguides for TM mode silicon photonics,” Opt. Express 23(19), 25295–25307 (2015).
[Crossref]

Cheben, P.

P. Cheben, J. Čtyroký, J. H. Schmid, S. Wang, J. Lapointe, J. G. Wangüemert-Pérez, Í Molina-Fernández, A. Ortega-Moñux, R. Halir, D. Melati, D. Xu, S. Janz, and M. Dado, “Bragg filter bandwidth engineering in subwavelength grating metamaterial waveguides,” Opt. Lett. 44(4), 1043–1046 (2019).
[Crossref]

J. Čtyroký, J. G. Wangüemert-Pérez, P. Kwiecien, I. Richter, J. Litvik, J. H. Schmid, Í Molina-Fernández, A. Ortega-Moñux, M. Dado, and P. Cheben, “Design of narrowband Bragg spectral filters in subwavelength grating metamaterial waveguides,” Opt. Express 26(1), 179–194 (2018).
[Crossref]

R. Halir, A. Ortega-Moñux, D. Benedikovic, G. Z. Mashanovich, J. G. Wangüemert-Pérez, J. H. Schmid, Í Molina-Fernández, and P. Cheben, “Subwavelength-Grating Metamaterial Structures for Silicon Photonic Devices,” Proc. IEEE 106(12), 2144–2157 (2018).
[Crossref]

P. Cheben, R. Halir, J. H. Schmid, H. A. Atwater, and D. R. Smith, “Subwavelength integrated photonics,” Nature 560(7720), 565–572 (2018).
[Crossref]

A. Ortega-Moñux, J. Čtyroký, P. Cheben, J. H. Schmid, S. Wang, Í Molina-Fernández, and R. Halir, “Disorder effects in subwavelength grating metamaterial waveguides,” Opt. Express 25(11), 12222–12236 (2017).
[Crossref]

J. D. Sarmiento-Merenguel, A. Ortega-Moñux, J.-M. Fédéli, J. G. Wangüemert-Pérez, C. Alonso-Ramos, E. Durán-Valdeiglesias, P. Cheben, Í Molina-Fernández, and R. Halir, “Controlling leakage losses in subwavelength grating silicon metamaterial waveguides,” Opt. Lett. 41(15), 3443–3446 (2016).
[Crossref]

R. Halir, P. J. Bock, P. Cheben, A. Ortega-Moñux, C. Alonso-Ramos, J. H. Schmid, J. Lapointe, D.-X. Xu, J. G. Wangüemert-Pérez, Í Molina-Fernández, and S. Janz, “Waveguide sub-wavelength structures: a review of principles and applications,” Laser Photonics Rev. 9(1), 25–49 (2015).
[Crossref]

P. Cheben, J. H. Schmid, S. Wang, D.-X. Xu, M. Vachon, S. Janz, J. Lapointe, Y. Painchaud, and M.-J. Picard, “Broadband polarization independent nanophotonic coupler for silicon waveguides with ultra-high efficiency,” Opt. Express 23(17), 22553–22563 (2015).
[Crossref]

J. H. Schmid, P. Cheben, P. J. Bock, R. Halir, J. Lapointe, S. Janz, A. Delâge, A. Densmore, J.-M. Fédeli, T. J. Hall, B. Lamontagne, R. Ma, I. Molina-Fernández, and D.-X. Xu, “Refractive Index Engineering With Subwavelength Gratings in Silicon Microphotonic Waveguides,” IEEE Photonics J. 3(3), 597–607 (2011).
[Crossref]

P. Cheben, P. J. Bock, J. H. Schmid, J. Lapointe, S. Janz, D.-X. Xu, A. Densmore, A. Delâge, B. Lamontagne, and T. J. Hall, “Refractive index engineering with subwavelength gratings for efficient microphotonic couplers and planar waveguide multiplexers,” Opt. Lett. 35(15), 2526–2528 (2010).
[Crossref]

P. J. Bock, P. Cheben, J. H. Schmid, A. Delâge, D.-X. Xu, S. Janz, and T. J. Hall, “Sub-wavelength grating mode transformers in silicon slab waveguides,” Opt. Express 17(21), 19120–19133 (2009).
[Crossref]

J. H. Schmid, P. Cheben, S. Janz, J. Lapointe, E. Post, and D.-X. Xu, “Gradient-index antireflective subwavelength structures for planar waveguide facets,” Opt. Lett. 32(13), 1794–1796 (2007).
[Crossref]

P. Cheben, D.-X. Xu, S. Janz, and A. Densmore, “Subwavelength waveguide grating for mode conversion and light coupling in integrated optics,” Opt. Express 14(11), 4695–4702 (2006).
[Crossref]

J. H. Schmid, P. Cheben, S. Janz, J. Lapointe, E. Post, A. Delâge, A. Densmore, B. Lamontagne, P. Waldron, and D. X. Xu, “Subwavelength Grating Structures in Silicon-on-Insulator waveguides,” Adv. Opt. Technol. 2008 (2008).

P. Cheben, S. Janz, D.-X. Xu, J. H. Schmid, A. Densmore, and J. Lapointe, “Interface device for performing mode transformation in optical waveguides,” U.S. patent 7,680,371 (16 March 2010).

P. Cheben, J. H. Schmid, D.-X. Xu, A. Densmore, and S. Janz, “Composite subwavelength-structured waveguide in optical systems,” U.S. patent 8,503,839 (6 August 2013).

Chen, C.-L.

C.-L. Chen, Foundations for Guided-Wave Optics (John Wiley & Sons, 2006).

Chen, L. R.

Chen, R. T.

Chen, Y.-J.

Y.-J. Hung, K.-H. Lin, C.-J. Wu, C.-Y. Wang, and Y.-J. Chen, “Narrowband Reflection From Weakly Coupled Cladding-Modulated Bragg Gratings,” IEEE J. Sel. Top. Quantum Electron. 22(6), 218–224 (2016).
[Crossref]

Chen, Z.

Cheng, R.

R. Cheng, N. A. F. Jaeger, and L. Chrostowski, “Fully tailorable integrated-optic resonators based on chirped waveguide Moiré gratings,” Optica 7(6), 647–657 (2020).
[Crossref]

R. Cheng, H. Yun, S. Lin, Y. Han, and L. Chrostowski, “Apodization profile amplification of silicon integrated Bragg gratings through lateral phase delays,” Opt. Lett. 44(2), 435–438 (2019).
[Crossref]

R. Cheng and L. Chrostowski, “Apodization of Silicon Integrated Bragg Gratings Through Periodic Phase Modulation,” IEEE J. Sel. Top. Quantum Electron. 25(5), 1–11 (2019).
[Crossref]

S. Kaushal, R. Cheng, M. Ma, A. Mistry, M. Burla, L. Chrostowski, and J. Azaña, “Optical signal processing based on silicon photonics waveguide Bragg gratings: review,” Front. Optoelectron. 11(2), 163–188 (2018).
[Crossref]

R. Cheng and L. Chrostowski, “Multichannel photonic Hilbert transformers based on complex modulated integrated Bragg gratings,” Opt. Lett. 43(5), 1031–1034 (2018).
[Crossref]

Chrostowski, L.

R. Cheng, N. A. F. Jaeger, and L. Chrostowski, “Fully tailorable integrated-optic resonators based on chirped waveguide Moiré gratings,” Optica 7(6), 647–657 (2020).
[Crossref]

R. Cheng, H. Yun, S. Lin, Y. Han, and L. Chrostowski, “Apodization profile amplification of silicon integrated Bragg gratings through lateral phase delays,” Opt. Lett. 44(2), 435–438 (2019).
[Crossref]

R. Cheng and L. Chrostowski, “Apodization of Silicon Integrated Bragg Gratings Through Periodic Phase Modulation,” IEEE J. Sel. Top. Quantum Electron. 25(5), 1–11 (2019).
[Crossref]

S. Kaushal, R. Cheng, M. Ma, A. Mistry, M. Burla, L. Chrostowski, and J. Azaña, “Optical signal processing based on silicon photonics waveguide Bragg gratings: review,” Front. Optoelectron. 11(2), 163–188 (2018).
[Crossref]

R. Cheng and L. Chrostowski, “Multichannel photonic Hilbert transformers based on complex modulated integrated Bragg gratings,” Opt. Lett. 43(5), 1031–1034 (2018).
[Crossref]

Z. Lu, J. Jhoja, J. Klein, X. Wang, A. Liu, J. Flueckiger, J. Pond, and L. Chrostowski, “Performance prediction for silicon photonics integrated circuits with layout-dependent correlated manufacturing variability,” Opt. Express 25(9), 9712–9733 (2017).
[Crossref]

H. P. Bazargani, M. Burla, L. Chrostowski, and J. Azaña, “Photonic Hilbert transformer based on laterally apodized integrated waveguide Bragg gratings on a SOI wafer,” Opt. Lett. 41(21), 5039–5042 (2016).
[Crossref]

M. Caverley, X. Wang, K. Murray, N. A. F. Jaeger, and L. Chrostowski, “Silicon-on-Insulator Modulators Using a Quarter-Wave Phase-Shifted Bragg Grating,” IEEE Photonics Technol. Lett. 27(22), 2331–2334 (2015).
[Crossref]

Z. Chen, J. Flueckiger, X. Wang, F. Zhang, H. Yun, Z. Lu, M. Caverley, Y. Wang, N. A. F. Jaeger, and L. Chrostowski, “Spiral Bragg grating waveguides for TM mode silicon photonics,” Opt. Express 23(19), 25295–25307 (2015).
[Crossref]

X. Wang, Y. Wang, J. Flueckiger, R. Bojko, A. Liu, A. Reid, J. Pond, N. A. F. Jaeger, and L. Chrostowski, “Precise control of the coupling coefficient through destructive interference in silicon waveguide Bragg gratings,” Opt. Lett. 39(19), 5519–5522 (2014).
[Crossref]

X. Wang, W. Shi, R. Vafaei, N. A. F. Jaeger, and L. Chrostowski, “Uniform and Sampled Bragg Gratings in SOI Strip Waveguides with Sidewall Corrugations,” IEEE Photonics Technol. Lett. 23(5), 290–292 (2011).
[Crossref]

H. Yun, M. Hammood, L. Chrostowski, and N. A. F. Jaeger, “Optical Add-drop Filters using Cladding-modulated Sub-wavelength Grating Contra-directional Couplers for Silicon-on-Insulator Platforms,” in IEEE 10th Annual Information Technology, Electronics and Mobile Communication Conference (IEMCON) (2019), pp. 0926–0931.

Ctyroký, J.

Dado, M.

Dagenais, M.

Dai, T.

Delâge, A.

J. H. Schmid, P. Cheben, P. J. Bock, R. Halir, J. Lapointe, S. Janz, A. Delâge, A. Densmore, J.-M. Fédeli, T. J. Hall, B. Lamontagne, R. Ma, I. Molina-Fernández, and D.-X. Xu, “Refractive Index Engineering With Subwavelength Gratings in Silicon Microphotonic Waveguides,” IEEE Photonics J. 3(3), 597–607 (2011).
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P. Cheben, P. J. Bock, J. H. Schmid, J. Lapointe, S. Janz, D.-X. Xu, A. Densmore, A. Delâge, B. Lamontagne, and T. J. Hall, “Refractive index engineering with subwavelength gratings for efficient microphotonic couplers and planar waveguide multiplexers,” Opt. Lett. 35(15), 2526–2528 (2010).
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P. J. Bock, P. Cheben, J. H. Schmid, A. Delâge, D.-X. Xu, S. Janz, and T. J. Hall, “Sub-wavelength grating mode transformers in silicon slab waveguides,” Opt. Express 17(21), 19120–19133 (2009).
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J. H. Schmid, P. Cheben, S. Janz, J. Lapointe, E. Post, A. Delâge, A. Densmore, B. Lamontagne, P. Waldron, and D. X. Xu, “Subwavelength Grating Structures in Silicon-on-Insulator waveguides,” Adv. Opt. Technol. 2008 (2008).

Densmore, A.

J. H. Schmid, P. Cheben, P. J. Bock, R. Halir, J. Lapointe, S. Janz, A. Delâge, A. Densmore, J.-M. Fédeli, T. J. Hall, B. Lamontagne, R. Ma, I. Molina-Fernández, and D.-X. Xu, “Refractive Index Engineering With Subwavelength Gratings in Silicon Microphotonic Waveguides,” IEEE Photonics J. 3(3), 597–607 (2011).
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P. Cheben, P. J. Bock, J. H. Schmid, J. Lapointe, S. Janz, D.-X. Xu, A. Densmore, A. Delâge, B. Lamontagne, and T. J. Hall, “Refractive index engineering with subwavelength gratings for efficient microphotonic couplers and planar waveguide multiplexers,” Opt. Lett. 35(15), 2526–2528 (2010).
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P. Cheben, D.-X. Xu, S. Janz, and A. Densmore, “Subwavelength waveguide grating for mode conversion and light coupling in integrated optics,” Opt. Express 14(11), 4695–4702 (2006).
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J. H. Schmid, P. Cheben, S. Janz, J. Lapointe, E. Post, A. Delâge, A. Densmore, B. Lamontagne, P. Waldron, and D. X. Xu, “Subwavelength Grating Structures in Silicon-on-Insulator waveguides,” Adv. Opt. Technol. 2008 (2008).

P. Cheben, J. H. Schmid, D.-X. Xu, A. Densmore, and S. Janz, “Composite subwavelength-structured waveguide in optical systems,” U.S. patent 8,503,839 (6 August 2013).

P. Cheben, S. Janz, D.-X. Xu, J. H. Schmid, A. Densmore, and J. Lapointe, “Interface device for performing mode transformation in optical waveguides,” U.S. patent 7,680,371 (16 March 2010).

Durán-Valdeiglesias, E.

Erdogan, T.

J. Skaar, L. Wang, and T. Erdogan, “On the Synthesis of Fiber Bragg Gratings by Layer Peeling,” IEEE J. Quantum Electron. 37(2), 165–173 (2001).
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Feced, R.

R. Feced, M. N. Zervas, and M. A. Muriel, “Efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings,” IEEE J. Quantum Electron. 35(8), 1105–1115 (1999).
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Fédeli, J.-M.

J. H. Schmid, P. Cheben, P. J. Bock, R. Halir, J. Lapointe, S. Janz, A. Delâge, A. Densmore, J.-M. Fédeli, T. J. Hall, B. Lamontagne, R. Ma, I. Molina-Fernández, and D.-X. Xu, “Refractive Index Engineering With Subwavelength Gratings in Silicon Microphotonic Waveguides,” IEEE Photonics J. 3(3), 597–607 (2011).
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Feng, C.

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Halir, R.

P. Cheben, J. Čtyroký, J. H. Schmid, S. Wang, J. Lapointe, J. G. Wangüemert-Pérez, Í Molina-Fernández, A. Ortega-Moñux, R. Halir, D. Melati, D. Xu, S. Janz, and M. Dado, “Bragg filter bandwidth engineering in subwavelength grating metamaterial waveguides,” Opt. Lett. 44(4), 1043–1046 (2019).
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P. Cheben, R. Halir, J. H. Schmid, H. A. Atwater, and D. R. Smith, “Subwavelength integrated photonics,” Nature 560(7720), 565–572 (2018).
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R. Halir, A. Ortega-Moñux, D. Benedikovic, G. Z. Mashanovich, J. G. Wangüemert-Pérez, J. H. Schmid, Í Molina-Fernández, and P. Cheben, “Subwavelength-Grating Metamaterial Structures for Silicon Photonic Devices,” Proc. IEEE 106(12), 2144–2157 (2018).
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A. Ortega-Moñux, J. Čtyroký, P. Cheben, J. H. Schmid, S. Wang, Í Molina-Fernández, and R. Halir, “Disorder effects in subwavelength grating metamaterial waveguides,” Opt. Express 25(11), 12222–12236 (2017).
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J. D. Sarmiento-Merenguel, A. Ortega-Moñux, J.-M. Fédéli, J. G. Wangüemert-Pérez, C. Alonso-Ramos, E. Durán-Valdeiglesias, P. Cheben, Í Molina-Fernández, and R. Halir, “Controlling leakage losses in subwavelength grating silicon metamaterial waveguides,” Opt. Lett. 41(15), 3443–3446 (2016).
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R. Halir, P. J. Bock, P. Cheben, A. Ortega-Moñux, C. Alonso-Ramos, J. H. Schmid, J. Lapointe, D.-X. Xu, J. G. Wangüemert-Pérez, Í Molina-Fernández, and S. Janz, “Waveguide sub-wavelength structures: a review of principles and applications,” Laser Photonics Rev. 9(1), 25–49 (2015).
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J. H. Schmid, P. Cheben, P. J. Bock, R. Halir, J. Lapointe, S. Janz, A. Delâge, A. Densmore, J.-M. Fédeli, T. J. Hall, B. Lamontagne, R. Ma, I. Molina-Fernández, and D.-X. Xu, “Refractive Index Engineering With Subwavelength Gratings in Silicon Microphotonic Waveguides,” IEEE Photonics J. 3(3), 597–607 (2011).
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Hall, T. J.

Hammood, M.

H. Yun, M. Hammood, L. Chrostowski, and N. A. F. Jaeger, “Optical Add-drop Filters using Cladding-modulated Sub-wavelength Grating Contra-directional Couplers for Silicon-on-Insulator Platforms,” in IEEE 10th Annual Information Technology, Electronics and Mobile Communication Conference (IEMCON) (2019), pp. 0926–0931.

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Hill, K. O.

K. O. Hill and G. Meltz, “Fiber Bragg Grating Technology Fundamentals and Overview,” J. Lightwave Technol. 15(8), 1263–1276 (1997).
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S. Xie, J. Zhan, Y. Hu, Y. Zhang, S. Veilleux, J. Bland-Hawthorn, and M. Dagenais, “Add-drop filter with complex waveguide Bragg grating and multimode interferometer operating on arbitrarily spaced channels,” Opt. Lett. 43(24), 6045–6048 (2018).
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Y.-J. Hung, K.-H. Lin, C.-J. Wu, C.-Y. Wang, and Y.-J. Chen, “Narrowband Reflection From Weakly Coupled Cladding-Modulated Bragg Gratings,” IEEE J. Sel. Top. Quantum Electron. 22(6), 218–224 (2016).
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Jaeger, N. A. F.

R. Cheng, N. A. F. Jaeger, and L. Chrostowski, “Fully tailorable integrated-optic resonators based on chirped waveguide Moiré gratings,” Optica 7(6), 647–657 (2020).
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M. Caverley, X. Wang, K. Murray, N. A. F. Jaeger, and L. Chrostowski, “Silicon-on-Insulator Modulators Using a Quarter-Wave Phase-Shifted Bragg Grating,” IEEE Photonics Technol. Lett. 27(22), 2331–2334 (2015).
[Crossref]

Z. Chen, J. Flueckiger, X. Wang, F. Zhang, H. Yun, Z. Lu, M. Caverley, Y. Wang, N. A. F. Jaeger, and L. Chrostowski, “Spiral Bragg grating waveguides for TM mode silicon photonics,” Opt. Express 23(19), 25295–25307 (2015).
[Crossref]

X. Wang, Y. Wang, J. Flueckiger, R. Bojko, A. Liu, A. Reid, J. Pond, N. A. F. Jaeger, and L. Chrostowski, “Precise control of the coupling coefficient through destructive interference in silicon waveguide Bragg gratings,” Opt. Lett. 39(19), 5519–5522 (2014).
[Crossref]

X. Wang, W. Shi, R. Vafaei, N. A. F. Jaeger, and L. Chrostowski, “Uniform and Sampled Bragg Gratings in SOI Strip Waveguides with Sidewall Corrugations,” IEEE Photonics Technol. Lett. 23(5), 290–292 (2011).
[Crossref]

H. Yun, M. Hammood, L. Chrostowski, and N. A. F. Jaeger, “Optical Add-drop Filters using Cladding-modulated Sub-wavelength Grating Contra-directional Couplers for Silicon-on-Insulator Platforms,” in IEEE 10th Annual Information Technology, Electronics and Mobile Communication Conference (IEMCON) (2019), pp. 0926–0931.

Janz, S.

P. Cheben, J. Čtyroký, J. H. Schmid, S. Wang, J. Lapointe, J. G. Wangüemert-Pérez, Í Molina-Fernández, A. Ortega-Moñux, R. Halir, D. Melati, D. Xu, S. Janz, and M. Dado, “Bragg filter bandwidth engineering in subwavelength grating metamaterial waveguides,” Opt. Lett. 44(4), 1043–1046 (2019).
[Crossref]

R. Halir, P. J. Bock, P. Cheben, A. Ortega-Moñux, C. Alonso-Ramos, J. H. Schmid, J. Lapointe, D.-X. Xu, J. G. Wangüemert-Pérez, Í Molina-Fernández, and S. Janz, “Waveguide sub-wavelength structures: a review of principles and applications,” Laser Photonics Rev. 9(1), 25–49 (2015).
[Crossref]

P. Cheben, J. H. Schmid, S. Wang, D.-X. Xu, M. Vachon, S. Janz, J. Lapointe, Y. Painchaud, and M.-J. Picard, “Broadband polarization independent nanophotonic coupler for silicon waveguides with ultra-high efficiency,” Opt. Express 23(17), 22553–22563 (2015).
[Crossref]

J. H. Schmid, P. Cheben, P. J. Bock, R. Halir, J. Lapointe, S. Janz, A. Delâge, A. Densmore, J.-M. Fédeli, T. J. Hall, B. Lamontagne, R. Ma, I. Molina-Fernández, and D.-X. Xu, “Refractive Index Engineering With Subwavelength Gratings in Silicon Microphotonic Waveguides,” IEEE Photonics J. 3(3), 597–607 (2011).
[Crossref]

P. Cheben, P. J. Bock, J. H. Schmid, J. Lapointe, S. Janz, D.-X. Xu, A. Densmore, A. Delâge, B. Lamontagne, and T. J. Hall, “Refractive index engineering with subwavelength gratings for efficient microphotonic couplers and planar waveguide multiplexers,” Opt. Lett. 35(15), 2526–2528 (2010).
[Crossref]

P. J. Bock, P. Cheben, J. H. Schmid, A. Delâge, D.-X. Xu, S. Janz, and T. J. Hall, “Sub-wavelength grating mode transformers in silicon slab waveguides,” Opt. Express 17(21), 19120–19133 (2009).
[Crossref]

J. H. Schmid, P. Cheben, S. Janz, J. Lapointe, E. Post, and D.-X. Xu, “Gradient-index antireflective subwavelength structures for planar waveguide facets,” Opt. Lett. 32(13), 1794–1796 (2007).
[Crossref]

P. Cheben, D.-X. Xu, S. Janz, and A. Densmore, “Subwavelength waveguide grating for mode conversion and light coupling in integrated optics,” Opt. Express 14(11), 4695–4702 (2006).
[Crossref]

J. H. Schmid, P. Cheben, S. Janz, J. Lapointe, E. Post, A. Delâge, A. Densmore, B. Lamontagne, P. Waldron, and D. X. Xu, “Subwavelength Grating Structures in Silicon-on-Insulator waveguides,” Adv. Opt. Technol. 2008 (2008).

P. Cheben, S. Janz, D.-X. Xu, J. H. Schmid, A. Densmore, and J. Lapointe, “Interface device for performing mode transformation in optical waveguides,” U.S. patent 7,680,371 (16 March 2010).

P. Cheben, J. H. Schmid, D.-X. Xu, A. Densmore, and S. Janz, “Composite subwavelength-structured waveguide in optical systems,” U.S. patent 8,503,839 (6 August 2013).

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Lamontagne, B.

J. H. Schmid, P. Cheben, P. J. Bock, R. Halir, J. Lapointe, S. Janz, A. Delâge, A. Densmore, J.-M. Fédeli, T. J. Hall, B. Lamontagne, R. Ma, I. Molina-Fernández, and D.-X. Xu, “Refractive Index Engineering With Subwavelength Gratings in Silicon Microphotonic Waveguides,” IEEE Photonics J. 3(3), 597–607 (2011).
[Crossref]

P. Cheben, P. J. Bock, J. H. Schmid, J. Lapointe, S. Janz, D.-X. Xu, A. Densmore, A. Delâge, B. Lamontagne, and T. J. Hall, “Refractive index engineering with subwavelength gratings for efficient microphotonic couplers and planar waveguide multiplexers,” Opt. Lett. 35(15), 2526–2528 (2010).
[Crossref]

J. H. Schmid, P. Cheben, S. Janz, J. Lapointe, E. Post, A. Delâge, A. Densmore, B. Lamontagne, P. Waldron, and D. X. Xu, “Subwavelength Grating Structures in Silicon-on-Insulator waveguides,” Adv. Opt. Technol. 2008 (2008).

Lapointe, J.

P. Cheben, J. Čtyroký, J. H. Schmid, S. Wang, J. Lapointe, J. G. Wangüemert-Pérez, Í Molina-Fernández, A. Ortega-Moñux, R. Halir, D. Melati, D. Xu, S. Janz, and M. Dado, “Bragg filter bandwidth engineering in subwavelength grating metamaterial waveguides,” Opt. Lett. 44(4), 1043–1046 (2019).
[Crossref]

R. Halir, P. J. Bock, P. Cheben, A. Ortega-Moñux, C. Alonso-Ramos, J. H. Schmid, J. Lapointe, D.-X. Xu, J. G. Wangüemert-Pérez, Í Molina-Fernández, and S. Janz, “Waveguide sub-wavelength structures: a review of principles and applications,” Laser Photonics Rev. 9(1), 25–49 (2015).
[Crossref]

P. Cheben, J. H. Schmid, S. Wang, D.-X. Xu, M. Vachon, S. Janz, J. Lapointe, Y. Painchaud, and M.-J. Picard, “Broadband polarization independent nanophotonic coupler for silicon waveguides with ultra-high efficiency,” Opt. Express 23(17), 22553–22563 (2015).
[Crossref]

J. H. Schmid, P. Cheben, P. J. Bock, R. Halir, J. Lapointe, S. Janz, A. Delâge, A. Densmore, J.-M. Fédeli, T. J. Hall, B. Lamontagne, R. Ma, I. Molina-Fernández, and D.-X. Xu, “Refractive Index Engineering With Subwavelength Gratings in Silicon Microphotonic Waveguides,” IEEE Photonics J. 3(3), 597–607 (2011).
[Crossref]

P. Cheben, P. J. Bock, J. H. Schmid, J. Lapointe, S. Janz, D.-X. Xu, A. Densmore, A. Delâge, B. Lamontagne, and T. J. Hall, “Refractive index engineering with subwavelength gratings for efficient microphotonic couplers and planar waveguide multiplexers,” Opt. Lett. 35(15), 2526–2528 (2010).
[Crossref]

J. H. Schmid, P. Cheben, S. Janz, J. Lapointe, E. Post, and D.-X. Xu, “Gradient-index antireflective subwavelength structures for planar waveguide facets,” Opt. Lett. 32(13), 1794–1796 (2007).
[Crossref]

J. H. Schmid, P. Cheben, S. Janz, J. Lapointe, E. Post, A. Delâge, A. Densmore, B. Lamontagne, P. Waldron, and D. X. Xu, “Subwavelength Grating Structures in Silicon-on-Insulator waveguides,” Adv. Opt. Technol. 2008 (2008).

P. Cheben, S. Janz, D.-X. Xu, J. H. Schmid, A. Densmore, and J. Lapointe, “Interface device for performing mode transformation in optical waveguides,” U.S. patent 7,680,371 (16 March 2010).

LaRochelle, S.

Le Roux, X.

Lin, K.-H.

Y.-J. Hung, K.-H. Lin, C.-J. Wu, C.-Y. Wang, and Y.-J. Chen, “Narrowband Reflection From Weakly Coupled Cladding-Modulated Bragg Gratings,” IEEE J. Sel. Top. Quantum Electron. 22(6), 218–224 (2016).
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S. Kaushal, R. Cheng, M. Ma, A. Mistry, M. Burla, L. Chrostowski, and J. Azaña, “Optical signal processing based on silicon photonics waveguide Bragg gratings: review,” Front. Optoelectron. 11(2), 163–188 (2018).
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J. H. Schmid, P. Cheben, P. J. Bock, R. Halir, J. Lapointe, S. Janz, A. Delâge, A. Densmore, J.-M. Fédeli, T. J. Hall, B. Lamontagne, R. Ma, I. Molina-Fernández, and D.-X. Xu, “Refractive Index Engineering With Subwavelength Gratings in Silicon Microphotonic Waveguides,” IEEE Photonics J. 3(3), 597–607 (2011).
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R. Halir, A. Ortega-Moñux, D. Benedikovic, G. Z. Mashanovich, J. G. Wangüemert-Pérez, J. H. Schmid, Í Molina-Fernández, and P. Cheben, “Subwavelength-Grating Metamaterial Structures for Silicon Photonic Devices,” Proc. IEEE 106(12), 2144–2157 (2018).
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Mazeas, F.

D. Oser, F. Mazeas, X. Le Roux, D. Pérez-Galacho, O. Alibart, S. Tanzilli, L. Labonté, D. Marris-Morini, L. Vivien, É. Cassan, and C. Alonso-Ramos, “Coherency-Broken Bragg Filters: Overcoming On-Chip Rejection Limitations,” Laser Photonics Rev. 13(8), 1800226 (2019).
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D. Pérez-Galacho, C. Alonso-Ramos, F. Mazeas, X. Le Roux, D. Oser, W. Zhang, D. Marris-Morini, L. Labonté, S. Tanzilli, É. Cassan, and L. Vivien, “Optical pump-rejection filter based on silicon sub-wavelength engineered photonic structures,” Opt. Lett. 42(8), 1468–1471 (2017).
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Melati, D.

Meltz, G.

K. O. Hill and G. Meltz, “Fiber Bragg Grating Technology Fundamentals and Overview,” J. Lightwave Technol. 15(8), 1263–1276 (1997).
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Meriggi, L.

Mistry, A.

S. Kaushal, R. Cheng, M. Ma, A. Mistry, M. Burla, L. Chrostowski, and J. Azaña, “Optical signal processing based on silicon photonics waveguide Bragg gratings: review,” Front. Optoelectron. 11(2), 163–188 (2018).
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Molina-Fernández, Í

P. Cheben, J. Čtyroký, J. H. Schmid, S. Wang, J. Lapointe, J. G. Wangüemert-Pérez, Í Molina-Fernández, A. Ortega-Moñux, R. Halir, D. Melati, D. Xu, S. Janz, and M. Dado, “Bragg filter bandwidth engineering in subwavelength grating metamaterial waveguides,” Opt. Lett. 44(4), 1043–1046 (2019).
[Crossref]

R. Halir, A. Ortega-Moñux, D. Benedikovic, G. Z. Mashanovich, J. G. Wangüemert-Pérez, J. H. Schmid, Í Molina-Fernández, and P. Cheben, “Subwavelength-Grating Metamaterial Structures for Silicon Photonic Devices,” Proc. IEEE 106(12), 2144–2157 (2018).
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J. Čtyroký, J. G. Wangüemert-Pérez, P. Kwiecien, I. Richter, J. Litvik, J. H. Schmid, Í Molina-Fernández, A. Ortega-Moñux, M. Dado, and P. Cheben, “Design of narrowband Bragg spectral filters in subwavelength grating metamaterial waveguides,” Opt. Express 26(1), 179–194 (2018).
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A. Ortega-Moñux, J. Čtyroký, P. Cheben, J. H. Schmid, S. Wang, Í Molina-Fernández, and R. Halir, “Disorder effects in subwavelength grating metamaterial waveguides,” Opt. Express 25(11), 12222–12236 (2017).
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J. D. Sarmiento-Merenguel, A. Ortega-Moñux, J.-M. Fédéli, J. G. Wangüemert-Pérez, C. Alonso-Ramos, E. Durán-Valdeiglesias, P. Cheben, Í Molina-Fernández, and R. Halir, “Controlling leakage losses in subwavelength grating silicon metamaterial waveguides,” Opt. Lett. 41(15), 3443–3446 (2016).
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R. Halir, P. J. Bock, P. Cheben, A. Ortega-Moñux, C. Alonso-Ramos, J. H. Schmid, J. Lapointe, D.-X. Xu, J. G. Wangüemert-Pérez, Í Molina-Fernández, and S. Janz, “Waveguide sub-wavelength structures: a review of principles and applications,” Laser Photonics Rev. 9(1), 25–49 (2015).
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Molina-Fernández, I.

J. H. Schmid, P. Cheben, P. J. Bock, R. Halir, J. Lapointe, S. Janz, A. Delâge, A. Densmore, J.-M. Fédeli, T. J. Hall, B. Lamontagne, R. Ma, I. Molina-Fernández, and D.-X. Xu, “Refractive Index Engineering With Subwavelength Gratings in Silicon Microphotonic Waveguides,” IEEE Photonics J. 3(3), 597–607 (2011).
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R. Feced, M. N. Zervas, and M. A. Muriel, “Efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings,” IEEE J. Quantum Electron. 35(8), 1105–1115 (1999).
[Crossref]

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M. Caverley, X. Wang, K. Murray, N. A. F. Jaeger, and L. Chrostowski, “Silicon-on-Insulator Modulators Using a Quarter-Wave Phase-Shifted Bragg Grating,” IEEE Photonics Technol. Lett. 27(22), 2331–2334 (2015).
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Figures (10)

Fig. 1.
Fig. 1. Schematic view of the cladding-modulated Bragg filter geometries analyzed in this paper: (a) continuous silicon waveguide core and (b) SWG metamaterial waveguide core (SiO2 upper cladding is not shown for clarity). Filter spectral response is controlled by modulating the separation ( ${s_n}$ ) between the waveguide core and the loading segments and the grating period ( ${\mathrm{\Lambda }_n}$ ). (c-d) Scanning electron microscope (SEM) images of a short section of the fabricated filters. A 2.2-µm upper SiO2 cladding was deposited by plasma-enhanced chemical vapor deposition after the SEM images were taken.
Fig. 2.
Fig. 2. Design flow to implement filters with an arbitrary frequency response. Part 1: Determination of the basic filter geometry and coupled-mode parameters. The waveguide core parameters are designed to achieve a low effective index and a correspondingly increased mode delocalization, and fulfill the Bragg condition at the central wavelength ${\lambda _0}$ . Then we calculate the group index ${n_{\textrm{g},\textrm{u}}}({{\lambda_0}} )$ , and the coupling $\kappa (s )$ and self-coupling $\kappa ^{\prime}(s )$ coefficients of the grating as a function of the separation s of the loading segments. Part 2: Calculation of the local reflection coefficients by using layer-peeling and layer-adding algorithms. The target spectrum in the wavelength domain, $r(\lambda )$ , is first converted into the wavenumber domain, $r(\delta )$ . The LPA returns the ideal local reflection coefficient profile ${\rho _n}$ . This profile is later shortened to minimize the filter length, then obtaining the final reflection coefficient profile ${\tilde{\rho }_n}$ to be implemented along the filter. The corresponding spectral response $\tilde{r}(\lambda )$ is calculated by using the LAA. Part 3: Filter apodization. The local reflectivity profile ${\tilde{\rho }_n}$ is synthesized period by period. The absolute value $|{{{\tilde{\rho }}_n}} |$ is implemented with the separation ${s_n}$ by using Eq. (8) and the mapping function $\kappa (s )$ . The phase $\angle {\tilde{\rho }_n}$ is synthesized by adjusting the period ${\mathrm{\Lambda }_n}$ according to Eq. (10) and the mapping function $\kappa ^{\prime}(s )$ .
Fig. 3.
Fig. 3. The filter specification: (a) target transmittance spectrum and (b) design parameters.
Fig. 4.
Fig. 4. (a) Effective index and (b) group index of the unperturbed Si-wire and SWG waveguides as a function of wavelength.
Fig. 5.
Fig. 5. Coupling coefficient distributions for (a) Si-wire and (b) SWG based filter designs.
Fig. 6.
Fig. 6. (a) Coupling and (b) self-coupling coefficients as a function of the separation s between the waveguide core and the loading segments, for the Si-wire and SWG waveguides.
Fig. 7.
Fig. 7. Final filter design: loading segment separation profile ${s_n}$ for the (a) Si-wire and (b) SWG filters; grating period profile ${\mathrm{\Lambda }_n}$ for the (c) Si-wire and (d) SWG filters.
Fig. 8.
Fig. 8. Final filter design: relative frequency distribution of the separations of the loading segments for the Si-wire (blue) and SWG (red) filters. Both histograms have been calculated using the same number of intervals.
Fig. 9.
Fig. 9. (a) Measured transmittances of the Si-wire filter (blue) and a Si-wire core waveguide without loading segments (black). (b) Measured transmittances of the SWG filter (red) and an SWG waveguide without loading segments (black). Simulated transmittance of the SWG waveguide in SOI with a 2 µm BOX (green). Losses of measurement setup, edge couplers and interconnecting waveguides have been subtracted from the experimental curves.
Fig. 10.
Fig. 10. Measured normalized transmittances of (a) the Si-wire filter (blue) and (b) the SWG filter (red). The corresponding target spectra are shown as black curves.

Tables (1)

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Table 1. Dimensional parameters of the Si-wire and SWG based filter designs.

Equations (10)

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Λ B = λ 0 2 n eff , u ( λ 0 ) .
δ ( λ ) = β u ( λ ) K B 2 2 π n g , u ( λ 0 ) λ λ 0 λ 0 2 ,
ρ n = Λ B π π / 2 Λ B π / 2 Λ B r n ( δ ) d δ .
r n + 1 ( δ ) = exp ( j 2 δ Λ B ) r n ( δ ) ρ n 1 ρ n r n ( δ ) .
ρ ~ n = ρ n + N 1 1 , 1 n N ~
r ~ n ( δ ) = r ~ n + 1 ( δ ) + ρ ~ n exp ( j 2 δ Λ B ) exp ( j 2 δ Λ B ) + ρ ~ n r ~ n + 1 ( δ ) , N ~ 1 n 1.
κ n = tan h 1 ( | ρ ~ n | ) Λ B .
2 Λ n β n ( λ 0 ) = 2 π + ρ ~ n + 1 ρ ~ n ,
β n ( λ 0 ) = β u ( λ 0 ) + κ n = π Λ B + κ n .
Λ n = Λ B ( 2 π + ρ ~ n + 1 ρ ~ n 2 π + 2 Λ B κ n ) .

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