Properties of reflection and transmission spectral filters based on Bragg gratings in subwavelength grating (SWG) metamaterial waveguides on silicon-on-insulator platform have been analyzed using proprietary 2D and 3D simulation tools based on Fourier modal method and the coupled-mode theory. We also demonstrate that the coupled Bloch mode theory can be advantageously applied to design of Bragg gratings in SWG waveguides. By combining different techniques, including judiciously positioning silicon loading segments within the evanescent field of the SWG waveguide and making use of its dispersion properties, it is possible to attain sub-nanometer spectral bandwidths for both reflection and transmission filters in the wavelength range of 1550 nm while keeping minimum structural features of the filters as large as 100 nm. Numerical simulations have also shown that a few nanometer jitter in the size and position of Si segments is well tolerated in our filter designs.
© 2018 Optical Society of America
Since the first demonstrations [1, 2], subwavelength grating waveguides (SWG) based on silicon-on-insulator (SOI) platform have become important building blocks in silicon photonics devices. SWG refractive index engineering brings unprecedented flexibility to the design of SOI waveguide components without excessive technological demands. Not only that most of such devices can be fabricated in single lithographic step but clever utilization of specific dispersion properties of SWG waveguides also allows for designing ultra-broadband devices. Among numerous applications of SWG technique, let us mention highly efficient fiber-chip couplers [1–6], broadband directional couplers  and multimode interference (MMI) couplers , polarization mode splitters [9–11], wavelength-division  and mode-division  multiplexers, optical delay lines , evanescent field sensors [14, 15] and suspended membrane waveguides for mid-infrared applications . Principles of operation and further applications of SWG devices have been recently reviewed in [17, 18].
In this paper, we present the first systematic study of narrowband reflection and transmission spectral filters in SOI SWG waveguides comprising Bragg gratings with lateral loading segments. First reports on Bragg gratings in silicon waveguides were published in early 2000s [19–21]. Since then, several papers were published on Bragg gratings based on longitudinally uniform rib or channel SOI waveguides [22–45], for applications including spectral filtering [23, 25, 33, 35, 46], sensing [24, 40] and optical signal processing [37, 43]. Design of Bragg grating in SWG waveguides was first reported only recently . By adjusting the effective waveguide core index, the mode size and profile can be tailored to optimize overlap with the Bragg grating structure. For example, by geometrically separating the Bragg grating from the waveguide core (so that it acts on the mode evanescent field) while at the same time delocalizing the waveguide mode by the SWG effect, the resulting interaction of the Bragg grating with the mode can be controlled more accurately compared to the conventional Bragg grating geometry. Furthermore, dispersion in SWG waveguides can be engineered , offering an additional degree of freedom in controlling the impulse response of the Bragg grating. In principle, Bragg grating in a SWG waveguide comprises two kinds of gratings, both of which are subwavelength, but with significantly different periods. While the SWG forming the waveguide is designed to operate outside its bandgap to ensure lossless propagation, the operation of the Bragg grating relies on its band gap. In , this condition was satisfied by choosing the period of the Bragg grating 2 times larger than that of the SWG waveguide. To this end, the authors proposed to change the length of each second Si segment of the original SWG grating. The structure is schematically shown in Fig. 1.
In , it was claimed that the bandwidth of the Bragg gratings with difference in lengths of about 10% between alternate silicon segments and with about 2000 grating periods is of the order of 1 nm. Our rigorous full-vector 3D simulations of the same structures predicted several times larger bandwidths . A detailed numerical analysis showed that even a small change of the position or width of Si segments (of the order of a few nanometers) induces a comparatively strong perturbation of the periodic structure, resulting in broadening of the Bragg resonance. This observation was also recently confirmed experimentally in our study of the effect of jitter in SWG waveguides .
These considerations initiated our systematic analysis of Bragg grating structures in SWG waveguides, as presented in this paper. We first consider several configurations of Bragg gratings, and from different possibilities we select two most promising implementations yielding high spectral selectivity. We also demonstrate that Bragg gratings in SWG waveguides can be described with good accuracy using the coupled mode theory (CMT), and for a particular geometry of the SWG waveguide we determine the necessary CMT parameters from results of rigorous 3D simulations. The CMT is then used for the design of apodized Bragg gratings with suppressed sidelobes. Finally, we propose and analyze narrowband and comb-like transmission filters based on Bragg gratings in SWG waveguides.
2. Design of Bragg gratings in SWG waveguides
Design of a Bragg grating typically starts from its required spectral bandwidth. Since the relative bandwidth of any grating device is ultimately determined by the number of “grooves” irradiated by the incident light, narrowband Bragg filter requires weak grating with large number of periods. The question of the “strength” of the Bragg grating in a SWG waveguide will be discussed in more detail in the next section. Generally, a weak grating implies that the Bloch mode propagating in the SWG waveguide is only weakly affected by the Bragg grating, irrespectively of the groove geometry. Bragg gratings in silicon waveguides are typically formed by surface [21, 26] or sidewall modulation [34, 36, 44–46]. Periodic arrays of cylinders placed aside the waveguide have also been used . The advantage of this technique is that locating these loading elements sufficiently far away from the waveguide core represents only weak modulation and thus enables to obtain reduced bandwidth without the need of shallow corrugation of the waveguide sidewalls. An interesting and simple method of fine tuning the coupling strength of the grating by relative shifting of periodic modulation at the left and right waveguide sidewalls has been presented in .
In this paper, we propose a new approach which, building upon these techniques, exploits the unique advantage of dispersion properties of SWG waveguides. For the first time, a Bragg grating comprising SWG metamaterial waveguide core loaded with lateral segments is proposed an analyzed. The structures are designed for SOI waveguides with silicon thickness of 220 nm and 3 micron buried oxide (BOX) layer. For simplicity, we follow the idea of [47, 48] that the Bragg grating period is 2 × the period of the SWG waveguide, albeit other schemes are also possible. This choice ensures that the input and output SWG waveguides operate reasonably far from their bandgap while the Bragg grating exhibits bandgap in the desired wavelength range, while the effective refractive indices of Bloch modes of both sections are above the limit required for negligible leakage to silicon substrate .
Based on these considerations, we choose the SWG metamaterial waveguide consisting of 400-nm-wide and 145-nm-long silicon segments with the period embedded in silica cladding. The structural dimensions are compatible with the 193 nm deep UV fabrication process. The SWG waveguide and calculated distributions of dominant field components of the fundamental Bloch mode at the wavelength of 1550 nm are shown in Fig. 2.
Eight different configurations of Bragg gratings shown in Fig. 3 were analyzed. In design 1, each second Si segment of the SWG waveguide is shifted by the same amount along the waveguide axis. In designs 2 and 3, the longitudinal and transversal dimensions of each second Si segment are modified, respectively. In designs 4 to 6, small Si blocks–loading segments–are inserted at different locations of the SWG waveguide. Finally, in design 7, larger loading segments are positioned along the SWG waveguide, while in design 8, positions of the segments placed at opposite sides of the waveguide are mutually shifted in the longitudinal direction. Our general design strategy is that by controlling the positions of loading segments within the local field of the Bloch mode, we can optimize the spectral properties of the Bragg gratings.
Spectral properties of all these configurations were numerically investigated using our proprietary 3D Fourier modal method (FMM) simulators [51–53]. As it follows from our previous discussion, designs 1-3 lead to rather strong modification of the original SWG waveguide, so that dimensional changes of the order of a few nanometers are required to get narrow (~1 nm) reflection spectrum. The situation is more relaxed for designs 4 to 6 due to smaller sizes of loading silicon segments: for narrowband reflection, the dimensions t and w of these segments are in the range of a few tens of nanometers. From the fabrication point of view, the most promising are designs 7 and 8 since even for narrowband operation, the minimum feature sizes can still be larger than 100 nm, i.e. compatible with 193 nm deep-UV lithography. For this reason, we choose design 7 as our nominal design and design 8 as its modification. In the next sections, spectral properties of these two configurations are analyzed in detail.
3. Theory of Bragg gratings in SWG metamaterial waveguides
Operation of Bragg gratings in optical waveguides can be described using the coupled-mode theory (CMT). Let us remind that properly formulated CMT, which takes into account all waveguide modes (including radiation modes), is rigorous [54, 55]. For weak gratings, it is often possible to use only the fundamental forward and backward propagating modes and neglect all other modes. Complex amplitudes and of the fundamental modes satisfy the set of coupled first-order differential equations
Here, the time harmonic dependence of the fields in the complex representation is supposed to be of the form (suppressed in the equations), is the phase detuning factor, is the propagation constant of the unperturbed waveguide without grating (here the SWG metamaterial waveguide), is the self-coupling coefficient representing the change of the propagation constant due to the presence of the grating, is the coupling coefficient which determines the strength of the grating, and is the grating constant. The coupling coefficients are usually calculated by overlap integrals of the mode field of the unperturbed waveguide with the perturbation of the permittivity distribution due to the presence of the grating. In the following, without loss of generality, we consider both and to be real and positive. As it is known in the coupled-mode theory, the solution of the set of Eqs. (1) with boundary conditions , where L is the length of the grating, gives the (power) modal reflectance of the grating in the formEq. (2) reduces to
It is also useful to determine the bandwidth between the first nulls, BWFN , which is determined according to (2) by condition . For small bandwidths it can be approximately expressed as
Although the CMT was originally derived for gratings on conventional (longitudinally homogeneous) waveguides, it can be equally well applied to gratings in SWG waveguides, providing we substitute the forward and backward propagating modes of the conventional waveguide with the Bloch modes of the SWG waveguide. Since the transverse mode field distribution of Bloch modes varies periodically along the SWG waveguide, the determination of the coupling constants from overlap integrals would be quite complex. Here we propose and demonstrate that this problem can be circumvented by determining the coupling constant directly from reflectance spectra of the gratings calculated by another, rigorous numerical method. Once the coupling constants are determined, the design of the Bragg grating can be optimized for required specifications using the simpler and more flexible CMT. This approach has been carefully tested on a 2D (planar) model of the SWG waveguide with Bragg gratings similar to our nominal design using the proprietary software FEXEN based on the Fourier modal method (FMM) . The coupling coefficient was determined from the reflectance maxima using Eq. (3) for different lengths (or number of periods) of the grating. Exponential dependence of the coupling coefficient on the separation of loading segments from the waveguide was found, as expected from basic physical considerations (evanescent field decay).
Encouraged with results of the 2D analysis, we then performed similar but computationally more intensive simulations using our 3D FMM tools. We choose the square loading segments of the size of 130 × 130 nm2, for compatibility with 193 nm DUV lithography. For our nominal design we calculated reflectance spectra of gratings for different separations of loading segments from the SWG core, specifically s = 150 nm, 250 nm, 350 nm, 450 nm, 750 nm and 1000 nm. For each value of separation s, the reflectance spectra were calculated for several gratings with different number of Bragg periods, i.e., of different lengths. Correspondingly, the gratings length varied approximately from 8 µm to 2000 µm. From the calculated results, and making use of Eq. (3), the coupling coefficient associated to each separation s was determined. For the CMT to be applicable also for gratings in SWG waveguides (which should not be taken as granted because of a high refractive index contrast of the SWG waveguide and a possible role of evanescent modes not taken into account in the CMT), the reflectance values for the same separation but different grating lengths must satisfy Eq. (3). As it is evident from a typical example in Fig. 4, this condition is clearly satisfied. For each s, the coupling coefficient is then chosen from the best fit of the corresponding dependence.
Our simulations showed that the wavelengths of maximum reflectance depend rather significantly on the separation This spectral shift, shown in Fig. 5(a), is a consequence of the self-coupling described by the coefficient in the CMT Eqs. (1). Its value can be obtained from the spectral position of maximum reflectance,Fig. 5(b). It is observed that, for both coupling coefficients, the dependence on the separation s can be fitted with good accuracy (except for separations smaller than ~200 nm) with exponential functions with decay factors:
We also examined the dependence of the coupling constant of the Bragg grating on the relative shift of the loading segments in the modified nominal design 8 in Fig. 3. In  it has been shown that for longitudinally uniform waveguides the coupling constant decays with increasing according to the cosine law,Fig. 6(a).
It is apparent that the coupling constant of the SWG Bragg grating with smaller separations s decays slightly faster with the shift than according to the cosine function, and approaches the cosine function for larger separations In Fig. 6(b) it is observed that for separations larger than ~200 nm, the wavelength of maximum reflectance is independent of the shift This is because, for larger separations, the self-coupling coefficient depends only on the separation and not on This behavior is a consequence of the transversal field distribution of the Bloch mode of the SWG waveguide shown in Fig. 2: at larger separations, the longitudinal variations of the Bloch mode field are comparatively weak. From Fig. 5 and Fig. 6 it is apparent that by judicious choice of the lateral and longitudinal positions of loading segments the coupling coefficients can be controlled in a wide range (several orders of magnitude). This needs to be carefully considered in the design of apodized Bragg gratings, as will be shown in Section 4.
Rather weak longitudinal variations of the Bloch mode field at larger separations invokes the idea that both coupling constants and might be rather insensitive to simultaneous longitudinal shifting of loading segments at both sides along the waveguide in the same direction. Our numerical simulations (not explicitly shown here for brevity) clearly confirmed this expectation. This is an important observation since it opens the possibility of chirping the Bragg grating on the SWG waveguide without changing its strength.
Figure 7(a) shows the reflectance bandwidth of the nominal Bragg grating as the function of the separation s and the grating length, determined as follows: We first calculated the wavelength dependence of the effective index of the Bloch mode of a SWG waveguide using the Fourier modal method and calculated the group effective index Then, for each value of separation s, the central wavelength was determined from the curve shown in Fig. 5(a). Finally, the BWFN bandwidth was calculated from Eq. (4) in which the value of the coupling constant was determined from its exponential dependence on s, see Fig. 5(b).
It is apparent that nominal SWG Bragg gratings longer than ~1 mm, created by loading segments separated by more than 500 nm from the SWG waveguide, yield sub-nanometer reflection bands. This is remarkable given all structural dimensions of the gratings are larger than 100 nm. This case is illustrated in Fig. 7(b), showing the reflectance spectrum of a very weakly modulated grating with 65536 Bragg periods calculated with the 3D FMM. Note that this value of separation s is just at the edge of simulation window shown in Fig. 2(c), where the mode field of the SWG waveguide is very weak. The resulting BWFN bandwidth is ~50 pm, which agrees well with that calculated from the CMT theory, see inset in Fig. 6. Similar bandwidth can also be obtained using modified grating with shifted loading segments. However, for the separation of the required shift is which is only by 4 nm smaller than
4. Apodization of SWG Bragg filters
The CMT approach is especially efficient for analysis and design of apodized Bragg filters. Already in the early paper  it was demonstrated that a quadratic apodization of the coupling coefficient results in a strong suppression of sidelobes. As an example we consider here a Bragg grating with a linear profile of the longitudinal shift (modified nominal design). According to Fig. 6, for the separations of loading segments larger than 350 nm, the coupling coefficient closely follows the cosine dependence on To be on the safe side, we choose the separation To get the maximum reflectance close to 100%, the grating was designed with 2500 periods (grating length 1210 µm). Longitudinal apodization profile of the shift and the corresponding profiles of the coupling coefficients and are shown in Fig. 8(a) and Fig. 8(b), respectively. The reflectance spectrum of the apodized grating was calculated by numerical integration of the Runge-Kutta equation , and it is shown in Fig. 8(c). Non-apodized (uniform) grating spectrum is also shown for the reference The separation of the uniform grating was adjusted so that its coupling coefficient equals the average value of the coupling coefficient of the apodized grating; this is obtained for As expected, the cosine (i.e., close to parabolic) apodization of the coupling coefficient results in strong suppression of sidelobes of the reflectance spectrum, as shown in Fig. 8(c). Spectral shift between the reflectance spectra of gratings with and without apodization arises as a consequence of the difference in the self-coupling coefficients for separations and
Essentially the same apodization profile of the coupling coefficient as in Fig. 8(b) can be obtained by longitudinal variation of the separation according toFig. 8(a). The resulting profile is plotted in Fig. 9(a).
For this type of apodization, the variation of results in corresponding variation of the self-coupling coefficient , as shown in Fig. 9(b). The calculated reflectance spectrum is shown in Fig. 9(c), together with the spectrum of a grating apodized by varying . While the sidelobes of the reflectance spectrum at longer wavelengths are strongly suppressed for the s-apodized grating, the sidelobes at the short-wavelength side are actually increased. This is a consequence of the apodization-induced variations of the self-coupling coefficient which effectively manifests itself as a grating chirp [59–61].
5. Narrow-band transmission SWG Bragg filters
Bragg gratings act essentially as narrow-band reflection filters. However, narrow-band transmission filters can also be implemented with Bragg gratings as Fabry-Perot (FP) filters schematically shown in Fig. 10.
From a large number of possible implementations we select here one typical example. In Fig. 11(a) we show transmittance and reflectance spectra of the distributed Bragg reflector (DBR) formed by the SWG Bragg grating with separation and 256 Bragg periods. In Fig. 11(b) it is shown the transmittance spectrum of a FP resonator created by two DBRs separated by 16 periods of a SWG waveguide. The calculated spectral width of the FP resonance is about 90 pm, which corresponds to the resonator Q-factor of 17500.
Very high refractive index contrast of SOI waveguides can also be advantageously used to make broadband Bragg mirrors with only a few grating periods. Such mirrors can be readily implemented in SWG waveguides even without silicon loading segments, e.g., by doubling the SWG period and the length of the SWG waveguide core segments. An example of a FP resonator with such mirrors is schematically shown in Fig. 12.
The reflectance spectra of Bragg mirrors created by doubling the period of the SWG waveguide are shown in Fig. 13(a) for various numbers of periods. The calculated reflectance spectrum of the semi-infinite grating is also shown for reference.
The reflectance bandwidth exceeds 120 nm. A part of the transmittance spectrum of a FP resonator with 2048 SWG periods (i.e., about 0.5 millimeter long) is shown in Fig. 13(b). The structure behaves as a comb filter with free spectral range ~1 nm. The apparent unevenness of the transmittance maxima is a consequence of a limited computational wavelength grid.
6. Influence of jitter
In our recent study  we theoretically and experimentally analyzed the influence of disorder (“jitter”) in the size and position of silicon segments due to fabrication imperfections on the functionality of SWG metamaterial waveguide devices. We have found that random variations in positions and dimensions of silicon segments larger than several nanometers can substantially deteriorate device performance. Since Bragg gratings rely on interference effect, they may be particularly sensitive to such defects. In order to get some insight into this problem we simulated the effect of random fluctuations of lengths of silicon segments in the nominal design of the narrowband Bragg grating with 2048 Bragg periods, with loading segments separated by , see Fig. 14(a). We also studied an example of a FP resonator with DBR mirrors shown on Fig. 14(b).
The lengths of all silicon segments (both of the SWG waveguide core and of the loading segments) were jittered using pseudorandom numbers with normal distribution and standard deviation Simulation results are shown in Fig. 14. Interestingly, it is observed that effect of jitter manifests itself mainly outside the stopbands of the Bragg gratings. As a result, although the characteristics of both devices are affected, their basic functionality–the existence of reflection and transmission peaks–remains preserved. This finding is encouraging for practical implementations of these devices.
We reported results of numerical simulations of reflection and transmission spectral filters based on Bragg gratings in subwavelength grating metamaterial waveguides. We demonstrated that filters with spectral bandwidths as small as a few tens of picometers can be implemented in silicon waveguides while keeping minimum structural dimensions compatible with deep-UV lithography (>100 nm). This is achieved by using a fundamental Bloch mode in the SWG metamaterial waveguide with delocalized electric field, weakly coupled with the lateral loading segments forming the Bragg grating. The Bloch field delocalization allows to substantially increase the distance between the loading segments and the SWG core, therefore relaxing requirement on the positioning accuracy. Furthermore, by judiciously shifting the positions of loading segments at both sides of the waveguide, the grating strength as well as its chirp is controlled. This allows to engineer the filter bandwidth and to shape its spectral and impulse response. We also showed that high refractive index contrast of silicon waveguides can be exploited to design comb-like transmission filters simply by doubling the period of the SWG waveguide, to form the Fabry-Perot resonator. An additional practical advantage of our design strategy is that only a single lithographic step and full etch of silicon waveguide layer is required. We have also shown, for the first time, that the efficient and widely used CMT formalism can be advantageously applied to Bragg gratings in SWG metamaterial waveguides, providing the coupling coefficients are determined by rigorous 3D tools such as FMM. Our results open exciting prospects for development of new types of spectral filters in silicon nanophotonic waveguides building upon the fundamental principle of metamaterial waveguide engineering.
Czech Science Foundation (GACR) (16-00329S); Ministerio de Economía y Competitividad, Programa Estatal de Investigación, Desarrollo e Innovación Orientada a los Retos de la Sociedad (cofinanciado FEDER), (TEC2016-80718-R); Universidad de Málaga.
References and links
1. 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] [PubMed]
2. 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] [PubMed]
3. 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] [PubMed]
4. D. Benedikovic, P. Cheben, J. H. Schmid, D.-X. Xu, B. Lamontagne, S. Wang, J. Lapointe, R. Halir, A. Ortega-Moñux, S. Janz, and M. Dado, “Subwavelength index engineered surface grating coupler with sub-decibel efficiency for 220-nm silicon-on-insulator waveguides,” Opt. Express 23(17), 22628–22635 (2015). [CrossRef] [PubMed]
5. D. Benedikovic, C. Alonso-Ramos, P. Cheben, J. H. Schmid, S. Wang, R. Halir, A. Ortega-Moñux, D.-X. Xu, L. Vivien, J. Lapointe, S. Janz, and M. Dado, “Single-etch subwavelength engineered fiber-chip grating couplers for 1.3 µm datacom wavelength band,” Opt. Express 24(12), 12893–12904 (2016). [CrossRef] [PubMed]
6. A. Sánchez-Postigo, J. Gonzalo Wangüemert-Pérez, J. M. Luque-González, Í. Molina-Fernández, P. Cheben, C. A. Alonso-Ramos, R. Halir, J. H. Schmid, and A. Ortega-Moñux, “Broadband fiber-chip zero-order surface grating coupler with 0.4 dB efficiency,” Opt. Lett. 41(13), 3013–3016 (2016). [CrossRef] [PubMed]
7. H. Yun, Y. Wang, F. Zhang, Z. Lu, S. Lin, L. Chrostowski, and N. A. Jaeger, “Broadband 2 × 2 adiabatic 3 dB coupler using silicon-on-insulator sub-wavelength grating waveguides,” Opt. Lett. 41(13), 3041–3044 (2016). [CrossRef] [PubMed]
8. R. Halir, P. Cheben, J. M. Luque-González, J. D. Sarmiento-Merenguel, J. H. Schmid, J. G. Wangüemert-Pérez, D. X. Xu, S. Wang, A. Ortega-Moñux, and I. Molina-Fernández, “Ultra-broadband nanophotonic beamsplitter using an anisotropic sub-wavelength metamaterial,” Laser Photonics Rev. 10(6), 1039–1046 (2016). [CrossRef]
9. Y. Xiong, J. G. Wangüemert-Pérez, D.-X. Xu, J. H. Schmid, P. Cheben, and W. N. Ye, “Polarization splitter and rotator with subwavelength grating for enhanced fabrication tolerance,” Opt. Lett. 39(24), 6931–6934 (2014). [CrossRef] [PubMed]
10. L. Liu, Q. Deng, and Z. Zhou, “Manipulation of beat length and wavelength dependence of a polarization beam splitter using a subwavelength grating,” Opt. Lett. 41(21), 5126–5129 (2016). [CrossRef] [PubMed]
11. Y. Xu and J. Xiao, “Ultracompact and high efficient silicon-based polarization splitter-rotator using a partially-etched subwavelength grating coupler,” Sci. Rep. 6(1), 27949 (2016). [CrossRef] [PubMed]
12. D. Pérez-Galacho, D. Marris-Morini, A. Ortega-Moñux, J. G. Wangüemert-Pérez, and L. Vivien, “Add/drop mode-division multiplexer based on a Mach-Zehnder interferometer and periodic waveguides,” IEEE Photonics J. 7(4), 7800907 (2015). [CrossRef]
13. J. Wang, R. Ashrafi, R. Adams, I. Glesk, I. Gasulla, J. Capmany, and L. R. Chen, “Subwavelength grating enabled on-chip ultra-compact optical true time delay line,” Sci. Rep. 6(1), 30235 (2016). [CrossRef] [PubMed]
14. J. Gonzalo Wangüemert-Pérez, P. Cheben, A. Ortega-Moñux, C. Alonso-Ramos, D. Pérez-Galacho, R. Halir, I. Molina-Fernández, D. X. Xu, and J. H. Schmid, “Evanescent field waveguide sensing with subwavelength grating structures in silicon-on-insulator,” Opt. Lett. 39(15), 4442–4445 (2014). [CrossRef] [PubMed]
15. J. Flueckiger, S. Schmidt, V. Donzella, A. Sherwali, D. M. Ratner, L. Chrostowski, and K. C. Cheung, “Sub-wavelength grating for enhanced ring resonator biosensor,” Opt. Express 24(14), 15672–15686 (2016). [CrossRef] [PubMed]
16. J. S. Penades, A. Ortega-Moñux, M. Nedeljkovic, J. G. Wangüemert-Pérez, R. Halir, A. Z. Khokhar, C. Alonso-Ramos, Z. Qu, I. Molina-Fernández, P. Cheben, and G. Z. Mashanovich, “Suspended silicon mid-infrared waveguide devices with subwavelength grating metamaterial cladding,” Opt. Express 24(20), 22908–22916 (2016). [CrossRef] [PubMed]
17. R. Halir, P. J. Bock, P. Cheben, A. Ortega-Moñux, C. Alonso-Ramos, J. H. Schmid, J. Lapointe, D.-X. Xu, J. G. Wanguemert-Pérez, I. Molina-Fernández, and S. Janz, “Waveguide sub-wavelength structures: a review of principles and applications,” Laser Photonics Rev. 9(1), 25–49 (2015). [CrossRef]
18. L. R. Chen, “Subwavelength grating waveguide devices in silicon-on-insulators for integrated microwave photonics (Invited Paper),” Chin. Opt. Lett. 15(1), 010004 (2017). [CrossRef]
19. T. Aalto, S. Yliniemi, P. Heimala, P. Pekko, J. Simonen, and M. Kuittinen, 19. T. Aalto, S. Yliniemi, P. Heimala, P. Pekko, J. Simonen, and M. Kuittinen, “Integrated Bragg gratings in silicon-on-insulator waveguides,” in Integrated Optics: Devices, Materials, and Technologies Vi, Y. S. Sidorin and A. Tervonen, eds., 117–124 (2002).
20. S. P. Chan, V. M. N. Passaro, S. T. Lim, C. E. Png, W. Headley, G. Masanovic, G. T. Reed, R. M. H. Atta, G. Ensell, and A. G. R. Evans, 20. S. P. Chan, V. M. N. Passaro, S. T. Lim, C. E. Png, W. Headley, G. Masanovic, G. T. Reed, R. M. H. Atta, G. Ensell, and A. G. R. Evans, “Characterisation of integrated Bragg gratings on silicon-on-insulator rib waveguides,” in Semiconductor Optoelectronic Devices for Lightwave Communication, J. Piprek, ed., 273–283 (2003).
21. T. E. Murphy, J. T. Hastings, and H. I. Smith, “Fabrication and characterization of narrow-band Bragg-reflection filters in silicon-on-insulator ridge waveguides,” J. Lightwave Technol. 19(12), 1938–1942 (2001). [CrossRef]
22. P. Dong and A. G. Kirk, “Second-order Bragg waveguide grating as a 1D photonic band gap structure in SOI waveguide,” in Applications of Photonic Technology, Closing the Gap between Theory, Development, and Application, Pt 1 and 2, J. C. Armitage, S. Fafard, R. A. Lessard, and G. A. Lampropoulos, eds., 589–597 (2004).
23. S. P. Chan, V. M. N. Passaro, G. Z. Mashanovich, G. Ensell, and G. T. Reed, “Third order Bragg grating filters in small SOI waveguides,” J. Eur. Opt. Soc. 2, 07029 (2007). [CrossRef]
24. V. M. N. Passaro, R. Loiacono, G. D’Amico, and F. De Leonardis, “Design of Bragg Grating Sensors Based on Submicrometer Optical Rib Waveguides in SOI,” IEEE Sens. J. 8(9), 1603–1611 (2008). [CrossRef]
25. P. M. Waugh, “First order Bragg grating filters in silicon on insulator waveguides,” in Photonic Fiber and Crystal Devices: Advances in Materials and Innovations in Device Applications II, SPIE 7056, S. Yin and R. Guo, eds., 70561T (2008).
26. I. Giuntoni, D. Stolarek, H. Richter, S. Marschmeyer, J. Bauer, A. Gajda, J. Bruns, B. Tillack, K. Petermann, and L. Zimmermann, “Deep-UV Technology for the Fabrication of Bragg Gratings on SOI Rib Waveguides,” IEEE Photonics Technol. Lett. 21(24), 1894–1896 (2009). [CrossRef]
27. S. Homampour, M. P. Bulk, P. E. Jessop, and A. P. Knights, “Thermal tuning of planar Bragg gratings in silicon-on-insulator rib waveguides,” in Physica Status Solidi C: Current Topics in Solid State Physics, Vol 6, Suppl 1, S. Kasap, R. DeCorby, H. Ruda, F. Hegmann, and R. Kashyap, eds., S240–S243 (2009)
29. J. Bauer, D. Stolarek, L. Zimmermann, I. Giuntoni, U. Haak, H. Richter, S. Marschmeyer, A. Gajda, J. Bruns, K. Petermann, and B. Tillack, 29. J. Bauer, D. Stolarek, L. Zimmermann, I. Giuntoni, U. Haak, H. Richter, S. Marschmeyer, A. Gajda, J. Bruns, K. Petermann, and B. Tillack, “Deep-UV KrF lithography for the fabrication of Bragg gratings on SOI rib waveguides,” in 26th European Mask and Lithography Conference, SPIE 7545, U. F. W. Behringer and W. Maurer, eds., UNSP 75450L (2010). [CrossRef]
30. T. Kita, R. Ishikawa, and H. Yamada, “Photonic band-gap anomaly in SOI photonic-wire Bragg-grating filter,” in 2010 7th IEEE International Conference on Group IV Photonics, 237–239 (2010). [CrossRef]
31. R. Loiacono, G. T. Reed, R. Gwilliam, G. Z. Mashanovich, L. O’Faolain, T. Krauss, G. Lulli, C. Jeynes, and R. Jones, “Germanium implanted Bragg gratings in silicon on insulator waveguides,” in Silicon Photonics V, SPIE 7606, J. A. Kubby and G. T. Reed, eds., UNSP 76060G (2010).
32. A. D. Simard, Y. Painchaud, and S. LaRochelle, “Integrated Bragg gratings in curved waveguides,” in Photonics Society, 2010 23rd Annual Meeting of the IEEE, IEEE, Denver, CO, USA, 726–727 (2010). [CrossRef]
33. 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] [PubMed]
34. G. M. Jiang, R. Y. Chen, Q. A. Zhou, J. Y. Yang, M. H. Wang, and X. Q. Jiang, “Slab-modulated sidewall Bragg gratings in silicon-on-insulator ridge waveguides,” IEEE Photonics Technol. Lett. 23(1), 6–8 (2011).
35. X. Wang, W. Shi, S. Grist, H. Yun, N. A. F. Jaeger, and L. Chrostowski, 35X. Wang, W. Shi, S. Grist, H. Yun, N. A. F. Jaeger, and L. Chrostowski, “Narrow-band transmission filter using phase-shifted Bragg gratings in SOI waveguide,” 2011 IEEE Photonics Conference, 869–870 (2011). [CrossRef]
36. 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).
37. I. Giuntoni, D. Stolarek, D. I. Kroushkov, J. Bruns, L. Zimmermann, B. Tillack, and K. Petermann, “Continuously tunable delay line based on SOI tapered Bragg gratings,” Opt. Express 20(10), 11241–11246 (2012). [CrossRef] [PubMed]
38. A. D. Simard, Y. Painchaud, and S. LaRochelle, “Small-footprint Integrated Bragg Gratings in SOI Spiral Waveguides,” in International Quantum Electronics Conference Lasers and Electro-Optics Europe, IEEE, Munich, Germany (2013). [CrossRef]
39. C. J. Chae, E. Skafidas, and D. Y. Choi, “Compact Bragg grating reflectors in silicon waveguides and their application to resonator filters,” 2014 Optical Fiber Communications Conference and Exhibition (2014). [CrossRef]
40. F. De Leonardis, C. E. Campanella, B. Troia, A. G. Perri, and V. M. N. Passaro, “Performance of SOI Bragg Grating Ring Resonator for Nonlinear Sensing Applications,” Sensors (Basel) 14(9), 16017–16034 (2014). [CrossRef] [PubMed]
41. M. Ma and L. R. Chen, 41M. Ma and L. R. Chen, “Towards an integrated SPM-based 2R regenerator in SOI with waveguides incorporating Bragg gratings,” 2014 IEEE Photonics Society Summer Topical Meeting Series, 210–211 (2014). [CrossRef]
42. H. Y. Qiu, P. Yu, Y. X. Su, and X. Q. Jiang, “Slab-modulated uniform and sampled Bragg gratings in SOI ridge waveguides,” in Nanophotonics and Micro/Nano Optics II, SPIE 9277, Z. Zhou and K. Wada, eds., UNSP 92771S (2014).
43. W. Shi, V. Veerasubramanian, D. Patel, and D. V. Plant, “Tunable nanophotonic delay lines using linearly chirped contradirectional couplers with uniform Bragg gratings,” Opt. Lett. 39(3), 701–703 (2014). [CrossRef] [PubMed]
44. 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]
45. 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] [PubMed]
46. 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] [PubMed]
48. J. Čtyroký, P. Kwiecien, J. Wang, I. Richter, I. Glesk, and L. Chen, 48. J. Čtyroký, P. Kwiecien, J. Wang, I. Richter, I. Glesk, and L. Chen, “Simulations of waveguide Bragg grating filters based on subwavelength grating waveguide,” in SPIE Optics and Optoelectronics 2015, Integrated Optics: Physics and Simulations II, SPIE, 9516, 95160M (2015).
49. 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] [PubMed]
50. 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] [PubMed]
51. J. Čtyroký and I. Richter, “Recent developments in Fourier modal methods for modeling guided-wave devices,” in ICO International Conference on Information Photonics (IEEE Photonics Society, 2011), pp. 12–15. [CrossRef]
52. J. Čtyroký, “3-D Bidirectional Propagation Algorithm Based on Fourier Series,” J. Lightwave Technol. 30(23), 3699–3708 (2012). [CrossRef]
53. B. Maes, J. Petráček, S. Burger, P. Kwiecien, J. Luksch, and I. Richter, “Simulations of high-Q optical nanocavities with a gradual 1D bandgap,” Opt. Express 21(6), 6794–6806 (2013). [CrossRef] [PubMed]
54. D. Marcuse, Theory of dielectric optical waveguides, 2nd ed. Academic, San Diego (1991).
56. D. C. Flanders, H. Kogelnik, R. V. Schmidt, and C. V. Shank, “Grating filters for thin-film optical waveguides,” Appl. Phys. Lett. 24(4), 194–196 (1974). [CrossRef]
57. L. Zavargo-Peche, A. Ortega-Moñux, J. G. Wangüemert-Pérez, and I. Molina-Fernández, “Fourier based combined techniques to design novel sub-wavelength optical integrated devices,” Prog. Electromagnetics Res. 123, 447–465 (2012). [CrossRef]
58. H. Kogelnik, “Filter response of nonuniform almost-periodic structures,” Bell Syst. Tech. J. 55(1), 109–126 (1976). [CrossRef]
59. V. Mizrahi and J. E. Sipe, “Optical properties of photosensitive fiber phase gratings,” J. Lightwave Technol. 11(10), 1513–1517 (1993). [CrossRef]
60. J. E. Sipe, L. Poladian, and C. M. de Sterke, “Propagation through nonuniform grating structures,” J. Opt. Soc. Am. A 11(4), 1307–1320 (1994). [CrossRef]
61. T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15(8), 1277–1284 (1997). [CrossRef]