Abstract

Bragg gratings operating in reflection are versatile filters that are an important building block of photonic circuits but, so far, their use has been limited due to the absence of CMOS compatible integrated circulators. In this paper, we propose to introduce two identical Bragg gratings in the arms of a Mach-Zehnder interferometer built with multimode interference 2 x 2 couplers to provide a reflective filter without circulator. We show that this structure has unique properties that significantly reduce phase noise distortions, avoid the need for thermal phase tuning, and make it compatible with complex apodization functions implemented through superposition apodization. We experimentally demonstrate several Bragg grating filters with high quality reflection spectra. For example, we successfully fabricated a 4 nm dispersion-less square-shaped filter having a sidelobe suppression ratio better than 15 dB and an in-band phase response with a group delay standard deviation of 2.0 ps. This result will enable the fabrication of grating based narrowband reflective filters having sharp spectral responses, which represents a major improvement in the filtering capability of the silicon platform.

© 2015 Optical Society of America

1. Introduction

Over the past twenty years, Fiber Bragg gratings have been used in a wide variety of applications and recently much effort have been devoted to implement these devices in silicon-on-insulator (SOI) to improve the filtering capability of this platform. In photonic circuits, Bragg gratings can be used in three different ways. Firstly, they act as simple reflectors in lasers resonators [1,2], for example in distributed Bragg reflector (DBR) lasers or distributed feedback (DFB) lasers. Secondly, these filters can be used in transmission [3] for gain equalization [4] or sensing [5], for example. Thirdly, Bragg gratings can be used in reflection for pulse shaping applications [6], chromatic dispersion compensation [7], add/drop filters [8] and so on. To use integrated Bragg gratings in the latter configuration, the reflected signal must be separated from the input signal by directing it to a different output port. For example, this function can be done off-chip using a circulator as schematically illustrated in Fig. 1(a).

 figure: Fig. 1

Fig. 1 Schematic of a Bragg grating filter operated in reflection with the reflected signal directed to an output port distinct from the input port using (a) a circulator, (b) a 3 dB coupler, (c) a grating-assisted contra-directional coupler, and (d) a Mach-Zehnder interferometer structure.

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Uniform [9,10], phase shifted [9,11] and superimposed [12] integrated Bragg gratings (IBGs) have already been demonstrated in SOI. Phase engineered filters have also been developed with bandwidths ranging from 20 nm to 100 nm [13,14]. However, most IBGs demonstrated in SOI [9–14] have the common characteristic of being relatively short (a few hundreds of microns), which limits the filtering capabilities of such devices. Longer IBGs are required to precisely shape the spectral characteristics of filters having narrow bandwidth, sharp passband, well-controlled dispersion, multi-channel responses, etc. This results from the Fourier transform-like relationship between the grating physical structure and its spectral response. Recently, high quality narrowband square-shaped amplitude IBGs having well-controlled phase responses were demonstrated [15,16]. This last achievement was made possible because wide multimode waveguides (1000 nm) were used instead of the usual singlemode waveguide (500 nm). This procedure reduces the phase noise [17,18], which allows implementation of the grating lengths needed for the fabrication of sharp-narrowband IBGs.

Despite these advances, the lack of a practical on-chip CMOS compatible circulator prevents the use of such devices within photonic integrated circuits (PICs) in SOI. This is due to the absence of a CMOS compatible magneto-optic material. A possible solution is to use a 50/50 coupler as shown on Fig. 1(b). However this configuration involves at least an extra 6 dB losses for the reflected signal since it passes through the coupler twice. Another approach already demonstrated in SOI is to insert the grating in a contradirectional coupler structure [19,20] as shown in Fig. 1(c). This last approach is however very sensitive to fabrication imperfections since it requires small sidewall corrugations on waveguides placed in close proximity, which is challenging with CMOS processes. A second difficulty lies with the implementation of the apodization profile. In grating-assisted contra-directional coupler structures, an apodization technique based on the precise tuning of the proximity of the two waveguides has been demonstrated for smooth apodization profiles [21] but this is not suitable for fully custom IBGs involving, for example, several lobes. Finally, as discussed previously and in more detail in [15–18], phase noise due to sidewall roughness is an important limitation for IBGs in SOI due to the high index contrast and the high mode intensity at the core boundary. Fortunately, this phase noise can be markedly reduced by using wider multimode waveguides and, by controlling the phase noise, the fabricated grating length can be increased. However, this phase-noise reduction technique is not compatible with grating-assisted couplers because the use of multimode waveguides will produce cross coupling between many modes and strongly degrade the device performance. Therefore, there is a need to find an integrated waveguide structure that will allow IBGs to operate in reflection while being compatible with versatile apodization techniques and with phase noise reduction strategies. The goal is to fabricate complex apodized IBG filters that are characterized by apodization profiles showing more elaborate features than the simple truncated, sampled or single lobe functions commonly used in integrated waveguides. These complex apodization profiles typically show several lobes, with well-defined relative amplitudes, and these profiles may or may not include phase shifts

To allow complex apodized IBGs to be used in reflection without circulators in SOI, we propose to fabricate two identical IBGs in the two arms of a Mach-Zehnder interferometer (MZI) as shown in Fig. 1(d). This interferometric structure has already been proposed in different platforms. For example, in fibers [22,23] it was used to demonstrate optical add-drop multiplexers [24]. Similarly, planar optical add-drop multiplexers as well as Hilbert transformers have been done in silica-on-silicon [25–28]. Finally, some modeling and experimental work was carried out in silicon [29–32], but so far the spectral response of the filters fabricated in SOI is too distorted to make them useful for system application [30]. In this paper, we propose a new design of a MZI structure with IBGs (MZI-IBG), shown in Fig. 2, that overcomes many of the shortcomings of the previous implementation in planar silicon waveguides. The structure is robust to fabrication errors, does not require phase tuning of the two arms, and can be implemented with multimode waveguides that reduce phase noise. Furthermore, this MZI-IBG is compatible with the superposition and phase-modulation apodization techniques discussed in [15,16,33] which allows fabricating totally custom, long and narrowband grating structures. With the proposed device configuration, these apodized IBGs can be incorporated in PICs without the use of optical circulators.

 figure: Fig. 2

Fig. 2 Schematic of a MZI-IBG made with MMIs and phase noise reduction waveguides

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In the ideal MZI-IBG case shown in Fig. 1(d), the couplers split the power equally into the two MZI arms, the two optical waves arrive in phase at the two identical IBGs, and the MZI arms have identical optical lengths. As a result, the reflection and transmission ports have the same complex spectral responses as a single IBG without additional loss. Unfortunately, this simple structure cannot be directly implemented in SOI due to the high level of sidewall roughness that perturbs the device response. More specifically, the coupling ratio of directional couplers becomes asymmetric, the optical path length of the connecting waveguides becomes mismatched and the IBGs will probably exhibit uneven phase and amplitude responses [17,18]. All these effects result in high excess loss and distorted spectral responses, which make these filters unusable.

In this paper, rather than using standard couplers, we build the MZI using multimode interferometers (MMIs), as shown in Fig. 2. MMI can be implemented with wider input/output waveguides, thereby reducing phase noise. Furthermore, the spacing between the MMI ports allow the IBGs to be placed directly at the output ports of the input MMI coupler, which alleviates phase mismatch between the reflected fields from the two IBGs. All these characteristics are discussed in detail in the paper, which is organized as follows. In section 2, we first derive the spectral response of the four ports of the MZI-IBG. We then examine the specific issues related to the implementation of this design in SOI. In section 3, we present experimental results of simple IBG structures, namely uniform and phase shifted IBGs. We show that the unique combination of properties of the proposed structure leads to high quality spectral filtering of these IBGs operated in reflection on-chip. Finally, we demonstrate the potential of this technology by designing and characterizing an elaborate square-shaped dispersion-less filter that would be appropriate, for example, for on-chip wavelength division multiplexing/demultiplexing of communication channels.

2. Modeling and design of MZI-IBG

In this section, we first present a simple model of the transfer function of the MZI-IBG that we subsequently use to examine the impact of fabrication errors that can lead to non-ideal characteristics of the constituent elements. We first consider that the couplers can be modeled by a general 2 x 2 matrix linking the coupler input (Ein) and output (Eout) port fields. We write

[Eout,1Eout,2]=[ABCD][Ein,1Ein,2]
Similarly, we consider that the fields at one extremity of a Bragg grating can be linked to the field at z0+Lg by the matrix
[Ez0+Ez0]=[1tr*t*rt1t*][Ez0+Lg+Ez0+Lg],
where Ez0+is the slowly varying forward propagating field amplitude, Ez0 is the slowly varying backward propagating field amplitude at position z0. Also, r, t and Lg refer respectively to the grating reflection and transmission coefficients and to the grating length. Thus, assuming that the light enters the MZI-IBG by only one of the four input ports as indicated in Fig. 1(d), the four port outputs are given by
E1=(A1A1r1+B1C1ei2Δϕinr2)ei2ϕinE2=(A1C1r1+C1D1ei2Δϕinr2)ei2ϕinE3=(A1A2t1+C1B2eiΔϕineiΔϕoutt2)eiϕineiϕoutE4=(A1C2t1+C1D2eiΔϕineiΔϕoutt2)eiϕineiϕout
where the indices of the grating reflection and transmission coefficients and those attached to the ABCD matrix elements refer respectively to the two numbered IBGs and 2 x 2 couplers as indicated in Fig. 1(d). The ϕin and ϕout terms refer to the common phase terms accumulated in both MZI arms respectively before and after the gratings and the terms Δϕin and Δϕout are the phase mismatch terms accumulated in the connecting waveguides between the couplers and the IBGs.

In the ideal situation, both IBGs are identical (i.e. r = r1 = r2 and t = t1 = t2), the lossless couplers split the power equally into the two branches (i.e. A = D = 1/√2 and B = C = i/√2) and no phase mismatch is accumulated in the connecting waveguides (i.e. Δϕin = Δϕout = 0). In this situation, aside from constant phase terms, port 2 and 4 are equal to the IBGs reflection and transmission coefficients and the two other ports do not output any optical power. However, when imperfections are present in the device (see below), optical power is coupled to port 1 and 3, which degrades the performance of the filter. As a result, we will refer to port 1 as the excess reflection loss port, port 2 as the reflection port, port 3 as the excess transmission loss port and port 4 as the transmission port. To properly optimize the device, each potential source of error needs to be analyzed and mitigated. We proceed by examining the impact of non-ideal component one at a time, neglecting the other types of errors. While Mechin et al. [34] performed such an analysis for fiber Bragg grating add/drop multiplexer, we here focus on the difficulties inherent to IBGs in SOI.

2.1 Unbalanced IBG spectral responses

We first consider the impact of IBGs with non-equal reflectivity. In this case, the couplers are considered as splitting the power equally into the two branches (A = D = 1/√2 and B = C = i/√2) and there is no phase mismatch present (Δϕin = Δϕout = 0). Equation (3) can now be written as

E1=12(r1r2)ei2ϕinE3=12(t1t2)eiϕineiϕoutE2=12(r1+r2)iei2ϕinE4=12(t1+t2)ieiϕineiϕout
which shows that, when the IBG spectral responses are not equal, optical power is now present at the output of port 1 and 3. Also, Eq. (4) shows that the spectral response at the reflection and transmission ports is the mean of the two IBG spectral responses, which can either reduce distortions or introduce resonances due to interference effect [17]. To reduce power loss at port 1 and 3 and distortion at port 2 and 4, any source of error in IBG spectral responses must be suppressed. Phase noise due to sidewall roughness is a major contributor to IBG spectral distortion. It has been shown that the use of wide multimode waveguides instead of the usual singlemode waveguide (500 nm) can reduce the phase noise present in the IBGs hence improving the matching of their spectral responses. When implementing this solution, care must be taken to excite only the fundamental mode of the multimode waveguide.

2.2 Un-ideal couplers

Once again, we consider only one source of error related to the couplers. Therefore, there is no phase mismatch (Δϕin = Δϕout = 0), the IBG spectral responses are identical (r = r1 = r2 and t = t1 = t2), and Eq. (3) becomes

E1=r(A1A1+B1C1)ei2ϕinE3=t(A1A2+C1B2)eiϕineiϕoutE2=r(A1C1+C1D1)ei2ϕinE4=t(A1C2+C1D2)eiϕineiϕout.
Two potential candidates are considered as 2 x 2 couplers: MMIs and directional couplers. For the latter one, considering two waveguides with identical propagation constant, the ABCD matrix can be written as
ABCD=[1KiKiK1K]
where K=sin2(πLc/zB) is the coupling coefficient, Lc is the coupler length and zB is the beating length. Thus, Eq. (5) becomes
E1=r(12K1)ei2ϕinE2=2rK11K1iei2ϕinE3=t(1K11K2K1K2)eiϕineiϕout.E4=t(1K1K2+K11K2)ieiϕineiϕout
As can be seen from the previous set of equations, the optical power output from port 1 and 3 are increasing when unbalanced couplers are used and this will create additional losses in port 2 and 4. Thus directional couplers are not the best solution since such uneven splitting is likely to result from fabrication imperfections. MMIs are a much better coupler choice because their performance is more robust to fabrication imperfections. Small length variations, compared to the ideal MMI length, will not modify significantly the coupling ratio [35]. Furthermore, MMIs can be designed to accommodate wide multimode waveguide outputs which are directly compatible with the phase noise reduction technique discussed in section 2.1.

2.3 Phase mismatch in the connecting waveguides

Finally, we consider the impact of phase mismatch in the MZI arms. In this case, the couplers have perfect splitting ratios (i.e. A = D = 1/√2 and B = C = i/√2) and that the grating spectral response are identical (r = r1 = r2 and t = t1 = t2). As a result, Eq. (3) is now

E1=r2(1ei2Δϕin)ei2ϕinirΔϕinei2ϕinE2=r2(1+ei2Δϕin)iei2ϕinr(1+iΔϕin)iei2ϕinE3=t2(1eiΔϕineiΔϕout)eiϕineiϕouttΔϕin+Δϕout2ieiϕineiϕout,E4=t2(1+eiΔϕineiΔϕout)ieiϕineiϕoutt(1+i(Δϕin+Δϕout2))ieiϕineiϕout
which shows that tuning the phase mismatch can be used to switch the optical power from port 2 to port 1 and from port 4 to port 3. However, if the phase mismatch is relatively small (as shown by the approximation of Eq. (8)) or not well controlled, some optical power will be transferred to the output of port 1 and 3 and the IBGs spectral response amplitude in port 2 and 4 will be reduced. In the fiber version of the MZI with Bragg gratings [22–24], the optical path length had to be UV trimmed to minimize the excess losses. However, in silicon, we want to avoid post-fabrication trimming or thermal tuning since the first approach does not allow mass production while the second one is power consuming. The standard deviation of the waveguide optical path length mismatch resulting from sidewall roughness increases with the propagation length (typically it is a square root function) [36]. Consequently, this effect will be reduced by using wider multimode waveguides to reduce phase noise and keeping the connecting waveguides as short as possible. Specifically, this means that there should not be any tapers or s-bent between the MMIs and the IBGs. Thus, the IBGs must be positioned directly at the output of the MMIs resulting in connecting waveguides about half a grating period long which should make the excess losses very small.

2.4 Proposed MZI-IBG structure

Figure 2 shows the schematic of the proposed MZI-IBG structure, with MMI and multimode waveguides, for phase noise reduction. The input port is composed of a singlemode waveguide, which is commonly used for optical routing. Then, an adiabatic taper is used to increase the waveguide width at the input of the MMI. The MMI structures are compatible with the wide input/output waveguides necessary for phase noise reduction. Furthermore, the spacing of the output ports allows the IBGs to be positioned directly at the output of the MMI without any s-bends or other waveguide routing. The outputs of the two gratings are similarly connected to another identical 50/50 MMI and the waveguides are tapered down to match the input optical routing waveguide width.

In the fabricated devices, the MMI input/output waveguide width is either 1.5 μm (uniform and phase shifted gratings in section 3) or 1.0 μm (square-shaped dispersion-less filter in section 4), while the width of the MMI section is 5 μm. The modal content of all these sections were simulated with a commercial mode solver (MODE solutions, Lumerical) while the propagation was done with an in-house software developed in Matlab. The designed MMIs length is 90 μm and has a simulated total loss of 0.1 dB. The input/output adiabatic tapers are 100 μm long linear tapers. The MZI-IBG layout was generated using an open-source GDS Matlab library [37]. The MZI-IBGs were fabricated in SOI with a 220 nm core and 2 μm buried oxide undercladding using electron-beam lithography at University of Washington. The optical characterization was achieved by coupling light to the chip with single-etched focusing grating couplers [38] and the complex spectral response was obtained using a commercial optical frequency domain reflectometer (OVA, Luna Technologies).

3. Simple IBGs

In this section, the proposed MZI-IBG structure is tested with simple uniform and phase shifted IBGs. The IBGs have a uniform period of 276 nm, a length of 720 μm, corrugations amplitude of 110 nm and are fabricated on 1.5 μm wide waveguides. Two grating structures were tested: one where the grating period was uniform and one with a centered π-phase shift.

The spectral response amplitude and group delay of the reflection and transmission ports of the uniform and phase shifted gratings are shown in Fig. 3 and Fig. 4 respectively. No curve smoothing was applied to the measured spectra in this paper. The experimental measurements are shown in red. The insertion losses and the grating coupler spectral response were removed using a second-order polynomial fit performed outside the grating bandgap. From the depth of the grating transmission measured at port 4 and shown in Fig. 3(c), and knowing the grating length, the coupling coefficient (κ) of both gratings was determined to be κ ~2850 m−1. This value was used to fit the grating spectral responses shown as black lines in Fig. 3 and Fig. 4. We see that the overlap between calculated and measured amplitude and phase are very good both for the uniform grating [Fig. 3] and for the phase shifted grating [Fig. 4]. These results indicate that the proposed MZI-IBG is robust to fabrication errors resulting in spectral responses that do not show signs of degradation caused by either waveguide imperfections, MMIs asymmetry and phase mismatch in the ultra-short connecting waveguides.

 figure: Fig. 3

Fig. 3 Amplitude ((a) and (c)) and group delay ((b) and (d)) of the reflection ((a) and (b)) and transmission ((c) and (d)) ports of a uniform MZI-IBG. Experimental measurement results are in red and the response simulated using the extracted value of the coupling coefficient is in black.

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 figure: Fig. 4

Fig. 4 Amplitude ((a) and (c)) and group delay ((b) and (d)) of the reflection ((a) and (b)) and transmission ((c) and (d)) ports of a phase shifted MZI-IBG. Experimental measurement results are in red and the response simulated using the extracted value of the coupling coefficient is in black.

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To further confirm this conclusion, the excess reflection and transmission loss ports of the uniform grating were characterized and are displayed in red in Fig. 5 (the same data is also shown in Fig. 6 to facilitate the analysis), after removal of the coupling losses. The reflection data had to be time filtered to remove the reflection from the input cleaved fiber as discussed in [39]. These results show that both the excess transmission [Fig. 5(a) and Fig. 6(a)] and reflection [Fig. 5(b) and Fig. 6(b)] loss ports have an amplitude 20 dB lower than the reflection ports. These results are particularly conclusive considering that this device is neither thermally tuned nor stabilized, and noting further that no post-fabrication trimming has been done.

 figure: Fig. 5

Fig. 5 Amplitude of the excess a) transmission and b) reflection losses of a uniform MZI-IBG (experimental results in red). The optimal fit (Δκ = 200 m−1, SRA1 = 8% and SRA2 = 4%) is shown in black. Also shown are simulations done with with the same Δκ when the MMIs are ideal (green) and when the SRAs are twice the value (blue) used for the optimal fit.

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 figure: Fig. 6

Fig. 6 Amplitude of the excess a) transmission and b) reflection loss of a uniform MZI-IBG (experimental results in red). The optimal fit (Δκ = 200 m−1, SRA1 = 8% and SRA2 = 4%) is shown in black. Also shown are simulations done when Δκ = 0 (blue) and 400 m−1 (green) respectively with the same SRAs.

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To evaluate the source of the excess loss, we used Eq. (3) to fit the measured data. The best fit was obtained using a coupling coefficient mismatch (Δκ) between the two gratings of 200 m−1 and by adding a splitting ratio asymmetry (SRA) of 8% and 4% for the input and output MMIs. This result is displayed in black in Fig. 5 and Fig. 6. This fit can be performed because the different sources of imperfection have distinct spectral signatures.

Figure 5 also shows the impact of the variation of the SRAs on the excess transmission and reflection losses. The values of the SRAs are used to fit the excess transmission loss spectrum width and to fit the shape of the main lobe in the reflection excess loss port. When both SRAs are null, the spectrum of the transmission excess loss port becomes narrower while a dip is present in the reflection loss main lobe [green in Fig. 5]. If the SRAs are too high, the level of the excess losses in both ports exceeds the experimental measurements as shown by the simulations done when the SRAs is twice the optimum value [blue in Fig. 5]. As a result, a reasonable fit can be easily obtained.

Similarly, Fig. 6 shows the impact of the variation of the coupling coefficient on the excess transmission and reflection losses. When Δκ is null [blue in Fig. 6], the transmission excess loss port spectrum has a significant dip at the Bragg wavelength. When Δκ is twice the optimal value [green in Fig. 6], this notch becomes a dual-lobe structure. As a result, the optimal Δκ value was found by matching the experimental transmission excess loss measurement by removing the notch while minimizing the apparition of a dual-lobe response.

Finally, the addition of a phase mismatch in the connecting waveguides as well as a mismatch in both Bragg wavelengths were also considered for this fit. However, both of these fabrication errors resulted in asymmetric spectra, both in the excess transmission and reflection loss ports. As a result, we concluded that these sources of errors were negligible in this case since the experimental measurements were fairly symmetrical. Finally, as shown by Eq. (3), the spectral signature of the excess loss ports depends on the specific grating spectral response. As a result, when analyzing these ports for grating structures other than uniform gratings, the behavior will be modified and therefore simulations have to be performed in each case.

4. High-quality square-shaped dispersion-less filter

In this section, we demonstrate the fabrication of a square-shaped dispersion-less filter with a 3 dB bandwidth of 4 nm. The local Bragg wavelength (λB) and coupling coefficient (κ) profiles are designed using an integral layer peeling algorithm [40] and are shown in Fig. 7. The well-known Born approximation that relates the Fourier transform of the spectral response of a weak grating to its phase and coupling coefficient profiles is not strictly valid in this case [41]. Figure 7(b) shows that, although the coupling coefficient profile resembles a sinc squared function, it is asymmetric. The grating phase, shown in Fig. 7(a) as the local Bragg grating wavelength, experiences π phase shifts at the zeros of the κ profile. The calculated spectral response for this design, shown in Fig. 8 (black line), corresponds well to a square-shaped dispersion-less filter.

 figure: Fig. 7

Fig. 7 Bragg wavelength (a) and coupling coefficient (b) profiles of the square-shaped dispersion-less filter with a 3 dB bandwidth of 4 nm.

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 figure: Fig. 8

Fig. 8 Amplitude ((a) and (c)) and group delay ((b) and (d)) of the reflection ((a) and (b)) and transmission ((c) and (d)) ports of a square-shaped dispersion-less MZI-IBG. Experimental results are shown in red and the design in black.

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Implementing this design, or any other grating designs with elaborate and custom spectral responses, is difficult in SOI because the strong overlap of the optical mode with the sidewalls leads to large coupling coefficients, even when the grating corrugation amplitude is limited to a few nanometers. To obtain good quality spectral responses from IBG filters, it is critical to avoid truncating the apodization profile that has to be precisely implemented down to low values of coupling coefficients, for example in order to define the successive lobes of the profile in Fig. 7 (b). When apodizing the grating by changing the corrugation amplitude, the precision of the coupling coefficient apodization profile is strongly limited by the resolution of the fabrication process. For weakly coupled gratings, the dimensions of these corrections even fall below the resolution limit of most fabrication processes. Recently, we showed that it is advantageous to implement apodization profiles by adjusting the relative phase of the two sidewall gratings while keeping the corrugation amplitude constant [33]. With this technique, we successfully demonstrated the implementation of a filter with a Gaussian apodization profile.

The square-shaped dispersion-less filter has a 1.5 mm grating length and was fabricated on a 1 μm wide waveguide with 100 nm corrugations. Figure 8 shows the spectral responses in reflection ((a) and (b)) and transmission ((c) and (d)) that were measured (red) and calculated (black) from the apodized grating design shown in Fig. 7. As shown in Fig. 8 (a), the bandpass property of the dispersion-less filter shows excellent agreement with the design. In Fig. 8(b), the experimentally measured group delay experiences only very small oscillations in the 3 dB bandwidth of the filter (illustrated by the blue lines). The transmission group delay shows a lot of ripples since the amplitude of the transmitted field is very small, near the detection noise floor. In reflection, the filter displays a flat-top 3-dB bandwidth of 4 nm, a sidelobe suppression ratio better than 15 dB and group delay ripples with an in-band standard deviation of 2.0 ps. To our knowledge, it is the first time that IBG with such complex apodization profile Bragg grating and operated in reflection has been implemented in SOI, because to achieve this performance the tails of the apodization profile have to be precisely reproduced in the IBG and phase noise has to be mitigated.

The transmission and reflection excess loss port spectra are shown in red in Fig. 9. This design was fabricated in a different run but at the same foundry than the uniform and phase shifted IBGs. We found an increase of the overall loss of about 0.25 dB in both the reflection (port 2) and transmission (port 4) ports compared to the previous run. The in-band excess reflection loss (port 1) and out-of-band transmission loss (port 3) are nearly 11 dB higher. All these values are compatible with an effective index mismatch (Δneff) between the waveguides containing the IBGs of about 1x10−4, which corresponds to a variation of approximately 1 nm of the average waveguide width. Simulations done with the error parameters found in the previous section (uniform IBGs) in addition to the waveguide width variation of 1 nm are shown in black in Fig. 9 where a good fit with the experimental measurement is observed. The 1 nm width variation is also in agreement with the 0.25 dB increase in the loss of the reflection and transmission ports. The fact that a high-quality square-shaped dispersion-less filter response is retrieved at the reflection port (port 2) demonstrates the robustness of this MZI-IBG configuration to fabrication errors.

 figure: Fig. 9

Fig. 9 Amplitude of the excess a) transmission and b) reflection loss of a square-shaped dispersion-less MZI-IBG. The experimental results are in red and the optimal fit (Δκ = 200 m−1, SRA1 = 8% and SRA2 = 4% and Δneff = 1x10−4) is shown in black.

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5. Conclusion

In conclusion, the proposed MZI-IBG structure allows the fabrication of PICs with complex IBGs operated in reflection without the need of circulators, making this type of filter a fully compatible CMOS component. A key aspect of this structure is the use of multimode interferometers (MMIs) and multimode waveguides that solve many issues. Firstly, it reduces the impact of sidewall roughness present in SOI, thereby reducing spectral distortions in IBGs. Secondly, MMIs also make the structure more robust to fabrication errors that could cause splitting ratio asymmetry. Thirdly, the compatibility of MMI with multimode waveguides placed directly at its outputs eliminates the optical length mismatch and reduces the excess loss. Finally, since this MZI-IBG is compatible with the superposition apodization technique, such device allows to fully exploiting the potential of IBGs for precisely tailored spectral filtering, which was not possible before. As a demonstration, we successfully fabricated a 4 nm square-shaped dispersion-less filter having sidelobe suppression ratio better than 15 dB and group delay ripples with a standard deviation of 2.0 ps. This work, enabling long and complex IBGs, should significantly improve the implementation of many innovative filters in the silicon platform.

Acknowledgments

The SPEED research project (Silicon Photonic Electrically Engineered Devices) is funded by NSERC (RDCPJ438811-12), PROMPT (PJT-2011-17), and TeraXion. We also acknowledge the contribution and technical support of CMC Microsystems. This work is part of the Canada Research Chair program in Advanced photonic technologies for communications (APTEC).

References and links

1. C. A. Barrios, V. R. Almeida, R. R. Panepucci, B. S. Schmidt, and M. Lipson, “Compact silicon tunable Fabry-Perot resonator with low power consumption,” IEEE Photon. Technol. Lett. 16(2), 506–508 (2004). [CrossRef]  

2. Y. Painchaud, M. Poulin, C. Latrasse, and M. Picard, “Bragg grating based Fabry-Perot filters for characterizing silicon-on-insulator waveguides,” in 2012 IEEE 9th Int. Conf. Group IV Photonics GFP, 180–182 (2012). [CrossRef]  

3. J. Skaar, “Synthesis of fiber Bragg gratings for use in transmission,” J. Opt. Soc. Am. A 18(3), 557–564 (2001). [CrossRef]  

4. M. Rochette, M. Guy, S. LaRochelle, J. Lauzon, and F. Trepanier, “Gain equalization of EDFA’s with Bragg gratings,” IEEE Photon. Technol. Lett. 11(5), 536–538 (1999). [CrossRef]  

5. W. W. Morey, G. Meltz, and W. H. Glenn, “Fiber optic bragg grating sensors,” Proc. SPIE 1169, 98 (1990).

6. J. Azaa, “Ultrafast analog all-optical signal processors based on fiber-grating devices,” IEEE Photonics J. 2(3), 359–386 (2010). [CrossRef]  

7. Y. Painchaud, M. Lapointe, F. Trepanier, R. L. Lachance, C. Paquet, and M. Guy, “Recent progress on FBG-based tunable dispersion compensators for 40 Gb/s applications,” in Opt. Fiber Commun. Conf. Opt. Soc. Am., 1–3 (2007). [CrossRef]  

8. M. N. Zervas and M. K. Durkin, “Physical insights into inverse-scattering profiles and symmetric dispersionless FBG designs,” Opt. Express 21(15), 17472–17483 (2013). [CrossRef]   [PubMed]  

9. Y. Painchaud, M. Poulin, C. Latrasse, N. Ayotte, M.-J. Picard, and M. Morin, “Bragg grating notch filters in silicon-on-insulator waveguides,” in Bragg Gratings Photosensitivity, Poling Glass Waveguides, BW2E.3, Optical Society of America (2012).

10. X. Wang, W. Shi, H. Yun, S. Grist, N. A. F. Jaeger, and L. Chrostowski, “Narrow-band waveguide Bragg gratings on SOI wafers with CMOS-compatible fabrication process,” Opt. Express 20(14), 15547–15558 (2012). [CrossRef]   [PubMed]  

11. K. A. Rutkowska, D. Duchesne, M. J. Strain, R. Morandotti, M. Sorel, and J. Azaña, “Ultrafast all-optical temporal differentiators based on CMOS-compatible integrated-waveguide Bragg gratings,” Opt. Express 19(20), 19514–19522 (2011). [CrossRef]   [PubMed]  

12. 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]   [PubMed]  

13. D. T. H. Tan, K. Ikeda, R. E. Saperstein, B. Slutsky, and Y. Fainman, “Chip-scale dispersion engineering using chirped vertical gratings,” Opt. Lett. 33(24), 3013–3015 (2008). [CrossRef]   [PubMed]  

14. G. F. R. Chen, T. Wang, C. Donnelly, and D. T. H. Tan, “Second and third order dispersion generation using nonlinearly chirped silicon waveguide gratings,” Opt. Express 21(24), 29223–29230 (2013). [CrossRef]   [PubMed]  

15. 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]   [PubMed]  

16. A. D. Simard and S. LaRochelle, “High Performance Narrow Bandpass Filters Based on Integrated Bragg Gratings in Silicon-on-Insulator,” in Opt. Fiber Commun. Conf. Opt. Soc. Am., Tu3A.2 (2015). [CrossRef]  

17. A. D. Simard, N. Ayotte, Y. Painchaud, S. Bedard, and S. LaRochelle, “Impact of sidewall roughness on integrated Bragg gratings,” J. Lightwave Technol. 29(24), 3693–3704 (2011). [CrossRef]  

18. A. D. Simard, G. Beaudin, V. Aimez, Y. Painchaud, and S. Larochelle, “Characterization and reduction of spectral distortions in silicon-on-insulator integrated Bragg gratings,” Opt. Express 21(20), 23145–23159 (2013). [CrossRef]   [PubMed]  

19. D. T. H. Tan, K. Ikeda, and Y. Fainman, “Coupled chirped vertical gratings for on-chip group velocity dispersion engineering,” Appl. Phys. Lett. 95(14), 141109 (2009). [CrossRef]  

20. W. Shi, X. Wang, W. Zhang, L. Chrostowski, and N. A. F. Jaeger, “Contradirectional couplers in silicon-on-insulator rib waveguides,” Opt. Lett. 36(20), 3999–4001 (2011). [CrossRef]   [PubMed]  

21. W. Shi, H. Yun, C. Lin, J. Flueckiger, N. A. F. Jaeger, and L. Chrostowski, “Coupler-apodized Bragg-grating add-drop filter,” Opt. Lett. 38(16), 3068–3070 (2013). [CrossRef]   [PubMed]  

22. D. C. Johnson, K. O. Hill, F. Bilodeau, and S. Faucher, “New design concept for a narrowband wavelength-selective optical tap and combiner,” Electron. Lett. 23(13), 668–669 (1987). [CrossRef]  

23. F. Bilodeau, D. C. Johnson, S. Thériault, B. Malo, J. Albert, and K. O. Hill, “An All-Fiber Dense-Wavelength-Division Multiplexer-Demultiplexer Using Photoimprinted Bragg Gratings,” IEEE Photon. Technol. Lett. 7(4), 388–390 (1995). [CrossRef]  

24. Y.-K. Chen, C.-H. Chang, Y.-L. Yang, I.-Y. Kuo, and T.-C. Liang, “Mach–Zehnder fiber-grating-based fixed and reconfigurable multichannel optical add-drop multiplexers for DWDM networks,” Opt. Commun. 169(1–6), 245–262 (1999). [CrossRef]  

25. J.-M. Jouanno and M. Kristensen, “Low crosstalk planar optical add-drop multiplexer fabricated with UV-induced Bragg gratings,” Electron. Lett. 33(25), 2120–2121 (1997). [CrossRef]  

26. J. Kim, G. Li, and K. A. Winick, “Design and fabrication of a glass waveguide optical add-drop multiplexer by use of an amorphous-silicon overlay distributed Bragg reflector,” Appl. Opt. 43(3), 671–677 (2004). [CrossRef]   [PubMed]  

27. C. Sima, J. C. Gates, H. L. Rogers, P. L. Mennea, C. Holmes, M. N. Zervas, and P. G. R. Smith, “Phase controlled integrated interferometric single-sideband filter based on planar Bragg gratings implementing photonic Hilbert transform,” Opt. Lett. 38(5), 727–729 (2013). [CrossRef]   [PubMed]  

28. C. Sima, J. C. Gates, C. Holmes, P. L. Mennea, M. N. Zervas, and P. G. R. Smith, “Terahertz bandwidth photonic Hilbert transformers based on synthesized planar Bragg grating fabrication,” Opt. Lett. 38(17), 3448–3451 (2013). [CrossRef]   [PubMed]  

29. Z. Zou, L. Zhou, X. Li, and J. Chen, “Channel-spacing tunable silicon comb filter using two linearly chirped Bragg gratings,” Opt. Express 22(16), 19513–19522 (2014). [CrossRef]   [PubMed]  

30. J. Wang, J. Flueckiger, L. Chrostowski, and L. R. Chen, “Bandpass Bragg grating transmission filter on silicon-on-insulator,” in Group IV Photonics GFP 2014 IEEE 11th Int. Conf. On, 79–80, IEEE (2014). [CrossRef]  

31. H. Okayama, Y. Onawa, D. Shimura, H. Yaegashi, and H. Sasaki, “Polarization rotation Bragg grating using Si wire waveguide with non-vertical sidewall,” Opt. Express 22(25), 31371–31378 (2014). [CrossRef]   [PubMed]  

32. J. B. Khurgin and P. A. Morton, “Linearized Bragg grating assisted electro-optic modulator,” Opt. Lett. 39(24), 6946–6949 (2014). [CrossRef]   [PubMed]  

33. 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]  

34. D. Mechin, P. Yvernault, L. Brilland, and D. Pureur, “Influence of Bragg gratings phase mismatch in a Mach-Zehnder-based add-drop multiplexer,” J. Lightwave Technol. 21(5), 1411–1416 (2003). [CrossRef]  

35. L. B. Soldano and E. C. M. Pennings, “Optical multi-mode interference devices based on self-imaging: principles and applications,” J. Lightwave Technol. 13(4), 615–627 (1995). [CrossRef]  

36. C. A. Mack, “Analytical expression for impact of linewidth roughness on critical dimension uniformity,” J. Micro/Nanolithogr., MEMS, MOEMS 13(2), 020501 (2014). [CrossRef]  

37. N. Ayotte and A. D. Simard, “Matlab GDS Photonics Toolbox,” <http://www.mathworks.com/ matlabcentral/fileexchange/file_infos/46827-nicolasayotte-matlabgdsphotonicstoolbox>.

38. Y. Wang, X. Wang, J. Flueckiger, H. Yun, W. Shi, R. Bojko, N. A. F. Jaeger, and L. Chrostowski, “Focusing sub-wavelength grating couplers with low back reflections for rapid prototyping of silicon photonic circuits,” Opt. Express 22(17), 20652–20662 (2014). [CrossRef]   [PubMed]  

39. A. D. Simard, Y. Painchaud, and S. LaRochelle, “Characterization of integrated Bragg grating profiles,” in Bragg Gratings Photosensit. Poling Glass Waveguides, BM3D.7 (2012). [CrossRef]  

40. A. Rosenthal and M. Horowitz, “Inverse scattering algorithm for reconstructing strongly reflecting fiber Bragg gratings,” IEEE J. Quantum Electron. 39(8), 1018–1026 (2003). [CrossRef]  

41. H. Kogelnik, “Filter response of nonuniform almost-periodic structures,” Bell Syst. Tech. J. 55(1), 109–126 (1976). [CrossRef]  

References

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  1. C. A. Barrios, V. R. Almeida, R. R. Panepucci, B. S. Schmidt, and M. Lipson, “Compact silicon tunable Fabry-Perot resonator with low power consumption,” IEEE Photon. Technol. Lett. 16(2), 506–508 (2004).
    [Crossref]
  2. Y. Painchaud, M. Poulin, C. Latrasse, and M. Picard, “Bragg grating based Fabry-Perot filters for characterizing silicon-on-insulator waveguides,” in 2012 IEEE 9th Int. Conf. Group IV Photonics GFP, 180–182 (2012).
    [Crossref]
  3. J. Skaar, “Synthesis of fiber Bragg gratings for use in transmission,” J. Opt. Soc. Am. A 18(3), 557–564 (2001).
    [Crossref]
  4. M. Rochette, M. Guy, S. LaRochelle, J. Lauzon, and F. Trepanier, “Gain equalization of EDFA’s with Bragg gratings,” IEEE Photon. Technol. Lett. 11(5), 536–538 (1999).
    [Crossref]
  5. W. W. Morey, G. Meltz, and W. H. Glenn, “Fiber optic bragg grating sensors,” Proc. SPIE 1169, 98 (1990).
  6. J. Azaa, “Ultrafast analog all-optical signal processors based on fiber-grating devices,” IEEE Photonics J. 2(3), 359–386 (2010).
    [Crossref]
  7. Y. Painchaud, M. Lapointe, F. Trepanier, R. L. Lachance, C. Paquet, and M. Guy, “Recent progress on FBG-based tunable dispersion compensators for 40 Gb/s applications,” in Opt. Fiber Commun. Conf. Opt. Soc. Am., 1–3 (2007).
    [Crossref]
  8. M. N. Zervas and M. K. Durkin, “Physical insights into inverse-scattering profiles and symmetric dispersionless FBG designs,” Opt. Express 21(15), 17472–17483 (2013).
    [Crossref] [PubMed]
  9. Y. Painchaud, M. Poulin, C. Latrasse, N. Ayotte, M.-J. Picard, and M. Morin, “Bragg grating notch filters in silicon-on-insulator waveguides,” in Bragg Gratings Photosensitivity, Poling Glass Waveguides, BW2E.3, Optical Society of America (2012).
  10. X. Wang, W. Shi, H. Yun, S. Grist, N. A. F. Jaeger, and L. Chrostowski, “Narrow-band waveguide Bragg gratings on SOI wafers with CMOS-compatible fabrication process,” Opt. Express 20(14), 15547–15558 (2012).
    [Crossref] [PubMed]
  11. K. A. Rutkowska, D. Duchesne, M. J. Strain, R. Morandotti, M. Sorel, and J. Azaña, “Ultrafast all-optical temporal differentiators based on CMOS-compatible integrated-waveguide Bragg gratings,” Opt. Express 19(20), 19514–19522 (2011).
    [Crossref] [PubMed]
  12. 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] [PubMed]
  13. D. T. H. Tan, K. Ikeda, R. E. Saperstein, B. Slutsky, and Y. Fainman, “Chip-scale dispersion engineering using chirped vertical gratings,” Opt. Lett. 33(24), 3013–3015 (2008).
    [Crossref] [PubMed]
  14. G. F. R. Chen, T. Wang, C. Donnelly, and D. T. H. Tan, “Second and third order dispersion generation using nonlinearly chirped silicon waveguide gratings,” Opt. Express 21(24), 29223–29230 (2013).
    [Crossref] [PubMed]
  15. 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] [PubMed]
  16. A. D. Simard and S. LaRochelle, “High Performance Narrow Bandpass Filters Based on Integrated Bragg Gratings in Silicon-on-Insulator,” in Opt. Fiber Commun. Conf. Opt. Soc. Am., Tu3A.2 (2015).
    [Crossref]
  17. A. D. Simard, N. Ayotte, Y. Painchaud, S. Bedard, and S. LaRochelle, “Impact of sidewall roughness on integrated Bragg gratings,” J. Lightwave Technol. 29(24), 3693–3704 (2011).
    [Crossref]
  18. A. D. Simard, G. Beaudin, V. Aimez, Y. Painchaud, and S. Larochelle, “Characterization and reduction of spectral distortions in silicon-on-insulator integrated Bragg gratings,” Opt. Express 21(20), 23145–23159 (2013).
    [Crossref] [PubMed]
  19. D. T. H. Tan, K. Ikeda, and Y. Fainman, “Coupled chirped vertical gratings for on-chip group velocity dispersion engineering,” Appl. Phys. Lett. 95(14), 141109 (2009).
    [Crossref]
  20. W. Shi, X. Wang, W. Zhang, L. Chrostowski, and N. A. F. Jaeger, “Contradirectional couplers in silicon-on-insulator rib waveguides,” Opt. Lett. 36(20), 3999–4001 (2011).
    [Crossref] [PubMed]
  21. W. Shi, H. Yun, C. Lin, J. Flueckiger, N. A. F. Jaeger, and L. Chrostowski, “Coupler-apodized Bragg-grating add-drop filter,” Opt. Lett. 38(16), 3068–3070 (2013).
    [Crossref] [PubMed]
  22. D. C. Johnson, K. O. Hill, F. Bilodeau, and S. Faucher, “New design concept for a narrowband wavelength-selective optical tap and combiner,” Electron. Lett. 23(13), 668–669 (1987).
    [Crossref]
  23. F. Bilodeau, D. C. Johnson, S. Thériault, B. Malo, J. Albert, and K. O. Hill, “An All-Fiber Dense-Wavelength-Division Multiplexer-Demultiplexer Using Photoimprinted Bragg Gratings,” IEEE Photon. Technol. Lett. 7(4), 388–390 (1995).
    [Crossref]
  24. Y.-K. Chen, C.-H. Chang, Y.-L. Yang, I.-Y. Kuo, and T.-C. Liang, “Mach–Zehnder fiber-grating-based fixed and reconfigurable multichannel optical add-drop multiplexers for DWDM networks,” Opt. Commun. 169(1–6), 245–262 (1999).
    [Crossref]
  25. J.-M. Jouanno and M. Kristensen, “Low crosstalk planar optical add-drop multiplexer fabricated with UV-induced Bragg gratings,” Electron. Lett. 33(25), 2120–2121 (1997).
    [Crossref]
  26. J. Kim, G. Li, and K. A. Winick, “Design and fabrication of a glass waveguide optical add-drop multiplexer by use of an amorphous-silicon overlay distributed Bragg reflector,” Appl. Opt. 43(3), 671–677 (2004).
    [Crossref] [PubMed]
  27. C. Sima, J. C. Gates, H. L. Rogers, P. L. Mennea, C. Holmes, M. N. Zervas, and P. G. R. Smith, “Phase controlled integrated interferometric single-sideband filter based on planar Bragg gratings implementing photonic Hilbert transform,” Opt. Lett. 38(5), 727–729 (2013).
    [Crossref] [PubMed]
  28. C. Sima, J. C. Gates, C. Holmes, P. L. Mennea, M. N. Zervas, and P. G. R. Smith, “Terahertz bandwidth photonic Hilbert transformers based on synthesized planar Bragg grating fabrication,” Opt. Lett. 38(17), 3448–3451 (2013).
    [Crossref] [PubMed]
  29. Z. Zou, L. Zhou, X. Li, and J. Chen, “Channel-spacing tunable silicon comb filter using two linearly chirped Bragg gratings,” Opt. Express 22(16), 19513–19522 (2014).
    [Crossref] [PubMed]
  30. J. Wang, J. Flueckiger, L. Chrostowski, and L. R. Chen, “Bandpass Bragg grating transmission filter on silicon-on-insulator,” in Group IV Photonics GFP 2014 IEEE 11th Int. Conf. On, 79–80, IEEE (2014).
    [Crossref]
  31. H. Okayama, Y. Onawa, D. Shimura, H. Yaegashi, and H. Sasaki, “Polarization rotation Bragg grating using Si wire waveguide with non-vertical sidewall,” Opt. Express 22(25), 31371–31378 (2014).
    [Crossref] [PubMed]
  32. J. B. Khurgin and P. A. Morton, “Linearized Bragg grating assisted electro-optic modulator,” Opt. Lett. 39(24), 6946–6949 (2014).
    [Crossref] [PubMed]
  33. 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]
  34. D. Mechin, P. Yvernault, L. Brilland, and D. Pureur, “Influence of Bragg gratings phase mismatch in a Mach-Zehnder-based add-drop multiplexer,” J. Lightwave Technol. 21(5), 1411–1416 (2003).
    [Crossref]
  35. L. B. Soldano and E. C. M. Pennings, “Optical multi-mode interference devices based on self-imaging: principles and applications,” J. Lightwave Technol. 13(4), 615–627 (1995).
    [Crossref]
  36. C. A. Mack, “Analytical expression for impact of linewidth roughness on critical dimension uniformity,” J. Micro/Nanolithogr., MEMS, MOEMS 13(2), 020501 (2014).
    [Crossref]
  37. N. Ayotte and A. D. Simard, “Matlab GDS Photonics Toolbox,” < http://www.mathworks.com/ matlabcentral/fileexchange/file_infos/46827-nicolasayotte-matlabgdsphotonicstoolbox >.
  38. Y. Wang, X. Wang, J. Flueckiger, H. Yun, W. Shi, R. Bojko, N. A. F. Jaeger, and L. Chrostowski, “Focusing sub-wavelength grating couplers with low back reflections for rapid prototyping of silicon photonic circuits,” Opt. Express 22(17), 20652–20662 (2014).
    [Crossref] [PubMed]
  39. A. D. Simard, Y. Painchaud, and S. LaRochelle, “Characterization of integrated Bragg grating profiles,” in Bragg Gratings Photosensit. Poling Glass Waveguides, BM3D.7 (2012).
    [Crossref]
  40. A. Rosenthal and M. Horowitz, “Inverse scattering algorithm for reconstructing strongly reflecting fiber Bragg gratings,” IEEE J. Quantum Electron. 39(8), 1018–1026 (2003).
    [Crossref]
  41. H. Kogelnik, “Filter response of nonuniform almost-periodic structures,” Bell Syst. Tech. J. 55(1), 109–126 (1976).
    [Crossref]

2015 (1)

2014 (6)

2013 (6)

2012 (2)

X. Wang, W. Shi, H. Yun, S. Grist, N. A. F. Jaeger, and L. Chrostowski, “Narrow-band waveguide Bragg gratings on SOI wafers with CMOS-compatible fabrication process,” Opt. Express 20(14), 15547–15558 (2012).
[Crossref] [PubMed]

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 (3)

2010 (1)

J. Azaa, “Ultrafast analog all-optical signal processors based on fiber-grating devices,” IEEE Photonics J. 2(3), 359–386 (2010).
[Crossref]

2009 (1)

D. T. H. Tan, K. Ikeda, and Y. Fainman, “Coupled chirped vertical gratings for on-chip group velocity dispersion engineering,” Appl. Phys. Lett. 95(14), 141109 (2009).
[Crossref]

2008 (1)

2004 (2)

C. A. Barrios, V. R. Almeida, R. R. Panepucci, B. S. Schmidt, and M. Lipson, “Compact silicon tunable Fabry-Perot resonator with low power consumption,” IEEE Photon. Technol. Lett. 16(2), 506–508 (2004).
[Crossref]

J. Kim, G. Li, and K. A. Winick, “Design and fabrication of a glass waveguide optical add-drop multiplexer by use of an amorphous-silicon overlay distributed Bragg reflector,” Appl. Opt. 43(3), 671–677 (2004).
[Crossref] [PubMed]

2003 (2)

A. Rosenthal and M. Horowitz, “Inverse scattering algorithm for reconstructing strongly reflecting fiber Bragg gratings,” IEEE J. Quantum Electron. 39(8), 1018–1026 (2003).
[Crossref]

D. Mechin, P. Yvernault, L. Brilland, and D. Pureur, “Influence of Bragg gratings phase mismatch in a Mach-Zehnder-based add-drop multiplexer,” J. Lightwave Technol. 21(5), 1411–1416 (2003).
[Crossref]

2001 (1)

1999 (2)

M. Rochette, M. Guy, S. LaRochelle, J. Lauzon, and F. Trepanier, “Gain equalization of EDFA’s with Bragg gratings,” IEEE Photon. Technol. Lett. 11(5), 536–538 (1999).
[Crossref]

Y.-K. Chen, C.-H. Chang, Y.-L. Yang, I.-Y. Kuo, and T.-C. Liang, “Mach–Zehnder fiber-grating-based fixed and reconfigurable multichannel optical add-drop multiplexers for DWDM networks,” Opt. Commun. 169(1–6), 245–262 (1999).
[Crossref]

1997 (1)

J.-M. Jouanno and M. Kristensen, “Low crosstalk planar optical add-drop multiplexer fabricated with UV-induced Bragg gratings,” Electron. Lett. 33(25), 2120–2121 (1997).
[Crossref]

1995 (2)

F. Bilodeau, D. C. Johnson, S. Thériault, B. Malo, J. Albert, and K. O. Hill, “An All-Fiber Dense-Wavelength-Division Multiplexer-Demultiplexer Using Photoimprinted Bragg Gratings,” IEEE Photon. Technol. Lett. 7(4), 388–390 (1995).
[Crossref]

L. B. Soldano and E. C. M. Pennings, “Optical multi-mode interference devices based on self-imaging: principles and applications,” J. Lightwave Technol. 13(4), 615–627 (1995).
[Crossref]

1990 (1)

W. W. Morey, G. Meltz, and W. H. Glenn, “Fiber optic bragg grating sensors,” Proc. SPIE 1169, 98 (1990).

1987 (1)

D. C. Johnson, K. O. Hill, F. Bilodeau, and S. Faucher, “New design concept for a narrowband wavelength-selective optical tap and combiner,” Electron. Lett. 23(13), 668–669 (1987).
[Crossref]

1976 (1)

H. Kogelnik, “Filter response of nonuniform almost-periodic structures,” Bell Syst. Tech. J. 55(1), 109–126 (1976).
[Crossref]

Aimez, V.

Albert, J.

F. Bilodeau, D. C. Johnson, S. Thériault, B. Malo, J. Albert, and K. O. Hill, “An All-Fiber Dense-Wavelength-Division Multiplexer-Demultiplexer Using Photoimprinted Bragg Gratings,” IEEE Photon. Technol. Lett. 7(4), 388–390 (1995).
[Crossref]

Almeida, V. R.

C. A. Barrios, V. R. Almeida, R. R. Panepucci, B. S. Schmidt, and M. Lipson, “Compact silicon tunable Fabry-Perot resonator with low power consumption,” IEEE Photon. Technol. Lett. 16(2), 506–508 (2004).
[Crossref]

Ayotte, N.

Azaa, J.

J. Azaa, “Ultrafast analog all-optical signal processors based on fiber-grating devices,” IEEE Photonics J. 2(3), 359–386 (2010).
[Crossref]

Azaña, J.

Barrios, C. A.

C. A. Barrios, V. R. Almeida, R. R. Panepucci, B. S. Schmidt, and M. Lipson, “Compact silicon tunable Fabry-Perot resonator with low power consumption,” IEEE Photon. Technol. Lett. 16(2), 506–508 (2004).
[Crossref]

Beaudin, G.

Bedard, S.

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]

Bilodeau, F.

F. Bilodeau, D. C. Johnson, S. Thériault, B. Malo, J. Albert, and K. O. Hill, “An All-Fiber Dense-Wavelength-Division Multiplexer-Demultiplexer Using Photoimprinted Bragg Gratings,” IEEE Photon. Technol. Lett. 7(4), 388–390 (1995).
[Crossref]

D. C. Johnson, K. O. Hill, F. Bilodeau, and S. Faucher, “New design concept for a narrowband wavelength-selective optical tap and combiner,” Electron. Lett. 23(13), 668–669 (1987).
[Crossref]

Bojko, R.

Brilland, L.

Chang, C.-H.

Y.-K. Chen, C.-H. Chang, Y.-L. Yang, I.-Y. Kuo, and T.-C. Liang, “Mach–Zehnder fiber-grating-based fixed and reconfigurable multichannel optical add-drop multiplexers for DWDM networks,” Opt. Commun. 169(1–6), 245–262 (1999).
[Crossref]

Chen, G. F. R.

Chen, J.

Chen, Y.-K.

Y.-K. Chen, C.-H. Chang, Y.-L. Yang, I.-Y. Kuo, and T.-C. Liang, “Mach–Zehnder fiber-grating-based fixed and reconfigurable multichannel optical add-drop multiplexers for DWDM networks,” Opt. Commun. 169(1–6), 245–262 (1999).
[Crossref]

Chrostowski, L.

Donnelly, C.

Duchesne, D.

Durkin, M. K.

Fainman, Y.

D. T. H. Tan, K. Ikeda, and Y. Fainman, “Coupled chirped vertical gratings for on-chip group velocity dispersion engineering,” Appl. Phys. Lett. 95(14), 141109 (2009).
[Crossref]

D. T. H. Tan, K. Ikeda, R. E. Saperstein, B. Slutsky, and Y. Fainman, “Chip-scale dispersion engineering using chirped vertical gratings,” Opt. Lett. 33(24), 3013–3015 (2008).
[Crossref] [PubMed]

Faucher, S.

D. C. Johnson, K. O. Hill, F. Bilodeau, and S. Faucher, “New design concept for a narrowband wavelength-selective optical tap and combiner,” Electron. Lett. 23(13), 668–669 (1987).
[Crossref]

Flueckiger, J.

Gates, J. C.

Glenn, W. H.

W. W. Morey, G. Meltz, and W. H. Glenn, “Fiber optic bragg grating sensors,” Proc. SPIE 1169, 98 (1990).

Grist, S.

Guy, M.

M. Rochette, M. Guy, S. LaRochelle, J. Lauzon, and F. Trepanier, “Gain equalization of EDFA’s with Bragg gratings,” IEEE Photon. Technol. Lett. 11(5), 536–538 (1999).
[Crossref]

Y. Painchaud, M. Lapointe, F. Trepanier, R. L. Lachance, C. Paquet, and M. Guy, “Recent progress on FBG-based tunable dispersion compensators for 40 Gb/s applications,” in Opt. Fiber Commun. Conf. Opt. Soc. Am., 1–3 (2007).
[Crossref]

Hill, K. O.

F. Bilodeau, D. C. Johnson, S. Thériault, B. Malo, J. Albert, and K. O. Hill, “An All-Fiber Dense-Wavelength-Division Multiplexer-Demultiplexer Using Photoimprinted Bragg Gratings,” IEEE Photon. Technol. Lett. 7(4), 388–390 (1995).
[Crossref]

D. C. Johnson, K. O. Hill, F. Bilodeau, and S. Faucher, “New design concept for a narrowband wavelength-selective optical tap and combiner,” Electron. Lett. 23(13), 668–669 (1987).
[Crossref]

Holmes, C.

Horowitz, M.

A. Rosenthal and M. Horowitz, “Inverse scattering algorithm for reconstructing strongly reflecting fiber Bragg gratings,” IEEE J. Quantum Electron. 39(8), 1018–1026 (2003).
[Crossref]

Ikeda, K.

D. T. H. Tan, K. Ikeda, and Y. Fainman, “Coupled chirped vertical gratings for on-chip group velocity dispersion engineering,” Appl. Phys. Lett. 95(14), 141109 (2009).
[Crossref]

D. T. H. Tan, K. Ikeda, R. E. Saperstein, B. Slutsky, and Y. Fainman, “Chip-scale dispersion engineering using chirped vertical gratings,” Opt. Lett. 33(24), 3013–3015 (2008).
[Crossref] [PubMed]

Jaeger, N. A. F.

Johnson, D. C.

F. Bilodeau, D. C. Johnson, S. Thériault, B. Malo, J. Albert, and K. O. Hill, “An All-Fiber Dense-Wavelength-Division Multiplexer-Demultiplexer Using Photoimprinted Bragg Gratings,” IEEE Photon. Technol. Lett. 7(4), 388–390 (1995).
[Crossref]

D. C. Johnson, K. O. Hill, F. Bilodeau, and S. Faucher, “New design concept for a narrowband wavelength-selective optical tap and combiner,” Electron. Lett. 23(13), 668–669 (1987).
[Crossref]

Jouanno, J.-M.

J.-M. Jouanno and M. Kristensen, “Low crosstalk planar optical add-drop multiplexer fabricated with UV-induced Bragg gratings,” Electron. Lett. 33(25), 2120–2121 (1997).
[Crossref]

Khurgin, J. B.

Kim, J.

Kogelnik, H.

H. Kogelnik, “Filter response of nonuniform almost-periodic structures,” Bell Syst. Tech. J. 55(1), 109–126 (1976).
[Crossref]

Kristensen, M.

J.-M. Jouanno and M. Kristensen, “Low crosstalk planar optical add-drop multiplexer fabricated with UV-induced Bragg gratings,” Electron. Lett. 33(25), 2120–2121 (1997).
[Crossref]

Kuo, I.-Y.

Y.-K. Chen, C.-H. Chang, Y.-L. Yang, I.-Y. Kuo, and T.-C. Liang, “Mach–Zehnder fiber-grating-based fixed and reconfigurable multichannel optical add-drop multiplexers for DWDM networks,” Opt. Commun. 169(1–6), 245–262 (1999).
[Crossref]

Lachance, R. L.

Y. Painchaud, M. Lapointe, F. Trepanier, R. L. Lachance, C. Paquet, and M. Guy, “Recent progress on FBG-based tunable dispersion compensators for 40 Gb/s applications,” in Opt. Fiber Commun. Conf. Opt. Soc. Am., 1–3 (2007).
[Crossref]

Lapointe, M.

Y. Painchaud, M. Lapointe, F. Trepanier, R. L. Lachance, C. Paquet, and M. Guy, “Recent progress on FBG-based tunable dispersion compensators for 40 Gb/s applications,” in Opt. Fiber Commun. Conf. Opt. Soc. Am., 1–3 (2007).
[Crossref]

LaRochelle, S.

Lauzon, J.

M. Rochette, M. Guy, S. LaRochelle, J. Lauzon, and F. Trepanier, “Gain equalization of EDFA’s with Bragg gratings,” IEEE Photon. Technol. Lett. 11(5), 536–538 (1999).
[Crossref]

Li, G.

Li, X.

Liang, T.-C.

Y.-K. Chen, C.-H. Chang, Y.-L. Yang, I.-Y. Kuo, and T.-C. Liang, “Mach–Zehnder fiber-grating-based fixed and reconfigurable multichannel optical add-drop multiplexers for DWDM networks,” Opt. Commun. 169(1–6), 245–262 (1999).
[Crossref]

Lin, C.

Lipson, M.

C. A. Barrios, V. R. Almeida, R. R. Panepucci, B. S. Schmidt, and M. Lipson, “Compact silicon tunable Fabry-Perot resonator with low power consumption,” IEEE Photon. Technol. Lett. 16(2), 506–508 (2004).
[Crossref]

MacIntyre, D. S.

Mack, C. A.

C. A. Mack, “Analytical expression for impact of linewidth roughness on critical dimension uniformity,” J. Micro/Nanolithogr., MEMS, MOEMS 13(2), 020501 (2014).
[Crossref]

Malo, B.

F. Bilodeau, D. C. Johnson, S. Thériault, B. Malo, J. Albert, and K. O. Hill, “An All-Fiber Dense-Wavelength-Division Multiplexer-Demultiplexer Using Photoimprinted Bragg Gratings,” IEEE Photon. Technol. Lett. 7(4), 388–390 (1995).
[Crossref]

Mechin, D.

Meltz, G.

W. W. Morey, G. Meltz, and W. H. Glenn, “Fiber optic bragg grating sensors,” Proc. SPIE 1169, 98 (1990).

Mennea, P. L.

Meriggi, L.

Morandotti, R.

Morey, W. W.

W. W. Morey, G. Meltz, and W. H. Glenn, “Fiber optic bragg grating sensors,” Proc. SPIE 1169, 98 (1990).

Morton, P. A.

Okayama, H.

Onawa, Y.

Painchaud, Y.

A. D. Simard, G. Beaudin, V. Aimez, Y. Painchaud, and S. Larochelle, “Characterization and reduction of spectral distortions in silicon-on-insulator integrated Bragg gratings,” Opt. Express 21(20), 23145–23159 (2013).
[Crossref] [PubMed]

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]

A. D. Simard, N. Ayotte, Y. Painchaud, S. Bedard, and S. LaRochelle, “Impact of sidewall roughness on integrated Bragg gratings,” J. Lightwave Technol. 29(24), 3693–3704 (2011).
[Crossref]

Y. Painchaud, M. Lapointe, F. Trepanier, R. L. Lachance, C. Paquet, and M. Guy, “Recent progress on FBG-based tunable dispersion compensators for 40 Gb/s applications,” in Opt. Fiber Commun. Conf. Opt. Soc. Am., 1–3 (2007).
[Crossref]

Panepucci, R. R.

C. A. Barrios, V. R. Almeida, R. R. Panepucci, B. S. Schmidt, and M. Lipson, “Compact silicon tunable Fabry-Perot resonator with low power consumption,” IEEE Photon. Technol. Lett. 16(2), 506–508 (2004).
[Crossref]

Paquet, C.

Y. Painchaud, M. Lapointe, F. Trepanier, R. L. Lachance, C. Paquet, and M. Guy, “Recent progress on FBG-based tunable dispersion compensators for 40 Gb/s applications,” in Opt. Fiber Commun. Conf. Opt. Soc. Am., 1–3 (2007).
[Crossref]

Pennings, E. C. M.

L. B. Soldano and E. C. M. Pennings, “Optical multi-mode interference devices based on self-imaging: principles and applications,” J. Lightwave Technol. 13(4), 615–627 (1995).
[Crossref]

Pureur, D.

Rochette, M.

M. Rochette, M. Guy, S. LaRochelle, J. Lauzon, and F. Trepanier, “Gain equalization of EDFA’s with Bragg gratings,” IEEE Photon. Technol. Lett. 11(5), 536–538 (1999).
[Crossref]

Rogers, H. L.

Rosenthal, A.

A. Rosenthal and M. Horowitz, “Inverse scattering algorithm for reconstructing strongly reflecting fiber Bragg gratings,” IEEE J. Quantum Electron. 39(8), 1018–1026 (2003).
[Crossref]

Rutkowska, K. A.

Saperstein, R. E.

Sasaki, H.

Schmidt, B. S.

C. A. Barrios, V. R. Almeida, R. R. Panepucci, B. S. Schmidt, and M. Lipson, “Compact silicon tunable Fabry-Perot resonator with low power consumption,” IEEE Photon. Technol. Lett. 16(2), 506–508 (2004).
[Crossref]

Shi, W.

Shimura, D.

Sima, C.

Simard, A. D.

Skaar, J.

Slutsky, B.

Smith, P. G. R.

Soldano, L. B.

L. B. Soldano and E. C. M. Pennings, “Optical multi-mode interference devices based on self-imaging: principles and applications,” J. Lightwave Technol. 13(4), 615–627 (1995).
[Crossref]

Sorel, M.

Strain, M. J.

Tan, D. T. H.

Thériault, S.

F. Bilodeau, D. C. Johnson, S. Thériault, B. Malo, J. Albert, and K. O. Hill, “An All-Fiber Dense-Wavelength-Division Multiplexer-Demultiplexer Using Photoimprinted Bragg Gratings,” IEEE Photon. Technol. Lett. 7(4), 388–390 (1995).
[Crossref]

Thoms, S.

Trepanier, F.

M. Rochette, M. Guy, S. LaRochelle, J. Lauzon, and F. Trepanier, “Gain equalization of EDFA’s with Bragg gratings,” IEEE Photon. Technol. Lett. 11(5), 536–538 (1999).
[Crossref]

Y. Painchaud, M. Lapointe, F. Trepanier, R. L. Lachance, C. Paquet, and M. Guy, “Recent progress on FBG-based tunable dispersion compensators for 40 Gb/s applications,” in Opt. Fiber Commun. Conf. Opt. Soc. Am., 1–3 (2007).
[Crossref]

Wang, T.

Wang, X.

Wang, Y.

Winick, K. A.

Yaegashi, H.

Yang, Y.-L.

Y.-K. Chen, C.-H. Chang, Y.-L. Yang, I.-Y. Kuo, and T.-C. Liang, “Mach–Zehnder fiber-grating-based fixed and reconfigurable multichannel optical add-drop multiplexers for DWDM networks,” Opt. Commun. 169(1–6), 245–262 (1999).
[Crossref]

Yun, H.

Yvernault, P.

Zervas, M. N.

Zhang, W.

Zhou, L.

Zou, Z.

Appl. Opt. (1)

Appl. Phys. Lett. (1)

D. T. H. Tan, K. Ikeda, and Y. Fainman, “Coupled chirped vertical gratings for on-chip group velocity dispersion engineering,” Appl. Phys. Lett. 95(14), 141109 (2009).
[Crossref]

Bell Syst. Tech. J. (1)

H. Kogelnik, “Filter response of nonuniform almost-periodic structures,” Bell Syst. Tech. J. 55(1), 109–126 (1976).
[Crossref]

Electron. Lett. (2)

D. C. Johnson, K. O. Hill, F. Bilodeau, and S. Faucher, “New design concept for a narrowband wavelength-selective optical tap and combiner,” Electron. Lett. 23(13), 668–669 (1987).
[Crossref]

J.-M. Jouanno and M. Kristensen, “Low crosstalk planar optical add-drop multiplexer fabricated with UV-induced Bragg gratings,” Electron. Lett. 33(25), 2120–2121 (1997).
[Crossref]

IEEE J. Quantum Electron. (1)

A. Rosenthal and M. Horowitz, “Inverse scattering algorithm for reconstructing strongly reflecting fiber Bragg gratings,” IEEE J. Quantum Electron. 39(8), 1018–1026 (2003).
[Crossref]

IEEE Photon. Technol. Lett. (3)

F. Bilodeau, D. C. Johnson, S. Thériault, B. Malo, J. Albert, and K. O. Hill, “An All-Fiber Dense-Wavelength-Division Multiplexer-Demultiplexer Using Photoimprinted Bragg Gratings,” IEEE Photon. Technol. Lett. 7(4), 388–390 (1995).
[Crossref]

C. A. Barrios, V. R. Almeida, R. R. Panepucci, B. S. Schmidt, and M. Lipson, “Compact silicon tunable Fabry-Perot resonator with low power consumption,” IEEE Photon. Technol. Lett. 16(2), 506–508 (2004).
[Crossref]

M. Rochette, M. Guy, S. LaRochelle, J. Lauzon, and F. Trepanier, “Gain equalization of EDFA’s with Bragg gratings,” IEEE Photon. Technol. Lett. 11(5), 536–538 (1999).
[Crossref]

IEEE Photonics J. (1)

J. Azaa, “Ultrafast analog all-optical signal processors based on fiber-grating devices,” IEEE Photonics J. 2(3), 359–386 (2010).
[Crossref]

IEEE Photonics Technol. Lett. (1)

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]

J. Lightwave Technol. (3)

J. Micro/Nanolithogr., MEMS, MOEMS (1)

C. A. Mack, “Analytical expression for impact of linewidth roughness on critical dimension uniformity,” J. Micro/Nanolithogr., MEMS, MOEMS 13(2), 020501 (2014).
[Crossref]

J. Opt. Soc. Am. A (1)

Opt. Commun. (1)

Y.-K. Chen, C.-H. Chang, Y.-L. Yang, I.-Y. Kuo, and T.-C. Liang, “Mach–Zehnder fiber-grating-based fixed and reconfigurable multichannel optical add-drop multiplexers for DWDM networks,” Opt. Commun. 169(1–6), 245–262 (1999).
[Crossref]

Opt. Express (8)

Z. Zou, L. Zhou, X. Li, and J. Chen, “Channel-spacing tunable silicon comb filter using two linearly chirped Bragg gratings,” Opt. Express 22(16), 19513–19522 (2014).
[Crossref] [PubMed]

H. Okayama, Y. Onawa, D. Shimura, H. Yaegashi, and H. Sasaki, “Polarization rotation Bragg grating using Si wire waveguide with non-vertical sidewall,” Opt. Express 22(25), 31371–31378 (2014).
[Crossref] [PubMed]

A. D. Simard, G. Beaudin, V. Aimez, Y. Painchaud, and S. Larochelle, “Characterization and reduction of spectral distortions in silicon-on-insulator integrated Bragg gratings,” Opt. Express 21(20), 23145–23159 (2013).
[Crossref] [PubMed]

G. F. R. Chen, T. Wang, C. Donnelly, and D. T. H. Tan, “Second and third order dispersion generation using nonlinearly chirped silicon waveguide gratings,” Opt. Express 21(24), 29223–29230 (2013).
[Crossref] [PubMed]

X. Wang, W. Shi, H. Yun, S. Grist, N. A. F. Jaeger, and L. Chrostowski, “Narrow-band waveguide Bragg gratings on SOI wafers with CMOS-compatible fabrication process,” Opt. Express 20(14), 15547–15558 (2012).
[Crossref] [PubMed]

K. A. Rutkowska, D. Duchesne, M. J. Strain, R. Morandotti, M. Sorel, and J. Azaña, “Ultrafast all-optical temporal differentiators based on CMOS-compatible integrated-waveguide Bragg gratings,” Opt. Express 19(20), 19514–19522 (2011).
[Crossref] [PubMed]

M. N. Zervas and M. K. Durkin, “Physical insights into inverse-scattering profiles and symmetric dispersionless FBG designs,” Opt. Express 21(15), 17472–17483 (2013).
[Crossref] [PubMed]

Y. Wang, X. Wang, J. Flueckiger, H. Yun, W. Shi, R. Bojko, N. A. F. Jaeger, and L. Chrostowski, “Focusing sub-wavelength grating couplers with low back reflections for rapid prototyping of silicon photonic circuits,” Opt. Express 22(17), 20652–20662 (2014).
[Crossref] [PubMed]

Opt. Lett. (8)

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] [PubMed]

D. T. H. Tan, K. Ikeda, R. E. Saperstein, B. Slutsky, and Y. Fainman, “Chip-scale dispersion engineering using chirped vertical gratings,” Opt. Lett. 33(24), 3013–3015 (2008).
[Crossref] [PubMed]

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] [PubMed]

J. B. Khurgin and P. A. Morton, “Linearized Bragg grating assisted electro-optic modulator,” Opt. Lett. 39(24), 6946–6949 (2014).
[Crossref] [PubMed]

W. Shi, X. Wang, W. Zhang, L. Chrostowski, and N. A. F. Jaeger, “Contradirectional couplers in silicon-on-insulator rib waveguides,” Opt. Lett. 36(20), 3999–4001 (2011).
[Crossref] [PubMed]

W. Shi, H. Yun, C. Lin, J. Flueckiger, N. A. F. Jaeger, and L. Chrostowski, “Coupler-apodized Bragg-grating add-drop filter,” Opt. Lett. 38(16), 3068–3070 (2013).
[Crossref] [PubMed]

C. Sima, J. C. Gates, H. L. Rogers, P. L. Mennea, C. Holmes, M. N. Zervas, and P. G. R. Smith, “Phase controlled integrated interferometric single-sideband filter based on planar Bragg gratings implementing photonic Hilbert transform,” Opt. Lett. 38(5), 727–729 (2013).
[Crossref] [PubMed]

C. Sima, J. C. Gates, C. Holmes, P. L. Mennea, M. N. Zervas, and P. G. R. Smith, “Terahertz bandwidth photonic Hilbert transformers based on synthesized planar Bragg grating fabrication,” Opt. Lett. 38(17), 3448–3451 (2013).
[Crossref] [PubMed]

Proc. SPIE (1)

W. W. Morey, G. Meltz, and W. H. Glenn, “Fiber optic bragg grating sensors,” Proc. SPIE 1169, 98 (1990).

Other (7)

Y. Painchaud, M. Poulin, C. Latrasse, and M. Picard, “Bragg grating based Fabry-Perot filters for characterizing silicon-on-insulator waveguides,” in 2012 IEEE 9th Int. Conf. Group IV Photonics GFP, 180–182 (2012).
[Crossref]

Y. Painchaud, M. Poulin, C. Latrasse, N. Ayotte, M.-J. Picard, and M. Morin, “Bragg grating notch filters in silicon-on-insulator waveguides,” in Bragg Gratings Photosensitivity, Poling Glass Waveguides, BW2E.3, Optical Society of America (2012).

Y. Painchaud, M. Lapointe, F. Trepanier, R. L. Lachance, C. Paquet, and M. Guy, “Recent progress on FBG-based tunable dispersion compensators for 40 Gb/s applications,” in Opt. Fiber Commun. Conf. Opt. Soc. Am., 1–3 (2007).
[Crossref]

A. D. Simard and S. LaRochelle, “High Performance Narrow Bandpass Filters Based on Integrated Bragg Gratings in Silicon-on-Insulator,” in Opt. Fiber Commun. Conf. Opt. Soc. Am., Tu3A.2 (2015).
[Crossref]

J. Wang, J. Flueckiger, L. Chrostowski, and L. R. Chen, “Bandpass Bragg grating transmission filter on silicon-on-insulator,” in Group IV Photonics GFP 2014 IEEE 11th Int. Conf. On, 79–80, IEEE (2014).
[Crossref]

N. Ayotte and A. D. Simard, “Matlab GDS Photonics Toolbox,” < http://www.mathworks.com/ matlabcentral/fileexchange/file_infos/46827-nicolasayotte-matlabgdsphotonicstoolbox >.

A. D. Simard, Y. Painchaud, and S. LaRochelle, “Characterization of integrated Bragg grating profiles,” in Bragg Gratings Photosensit. Poling Glass Waveguides, BM3D.7 (2012).
[Crossref]

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Figures (9)

Fig. 1
Fig. 1 Schematic of a Bragg grating filter operated in reflection with the reflected signal directed to an output port distinct from the input port using (a) a circulator, (b) a 3 dB coupler, (c) a grating-assisted contra-directional coupler, and (d) a Mach-Zehnder interferometer structure.
Fig. 2
Fig. 2 Schematic of a MZI-IBG made with MMIs and phase noise reduction waveguides
Fig. 3
Fig. 3 Amplitude ((a) and (c)) and group delay ((b) and (d)) of the reflection ((a) and (b)) and transmission ((c) and (d)) ports of a uniform MZI-IBG. Experimental measurement results are in red and the response simulated using the extracted value of the coupling coefficient is in black.
Fig. 4
Fig. 4 Amplitude ((a) and (c)) and group delay ((b) and (d)) of the reflection ((a) and (b)) and transmission ((c) and (d)) ports of a phase shifted MZI-IBG. Experimental measurement results are in red and the response simulated using the extracted value of the coupling coefficient is in black.
Fig. 5
Fig. 5 Amplitude of the excess a) transmission and b) reflection losses of a uniform MZI-IBG (experimental results in red). The optimal fit (Δκ = 200 m−1, SRA1 = 8% and SRA2 = 4%) is shown in black. Also shown are simulations done with with the same Δκ when the MMIs are ideal (green) and when the SRAs are twice the value (blue) used for the optimal fit.
Fig. 6
Fig. 6 Amplitude of the excess a) transmission and b) reflection loss of a uniform MZI-IBG (experimental results in red). The optimal fit (Δκ = 200 m−1, SRA1 = 8% and SRA2 = 4%) is shown in black. Also shown are simulations done when Δκ = 0 (blue) and 400 m−1 (green) respectively with the same SRAs.
Fig. 7
Fig. 7 Bragg wavelength (a) and coupling coefficient (b) profiles of the square-shaped dispersion-less filter with a 3 dB bandwidth of 4 nm.
Fig. 8
Fig. 8 Amplitude ((a) and (c)) and group delay ((b) and (d)) of the reflection ((a) and (b)) and transmission ((c) and (d)) ports of a square-shaped dispersion-less MZI-IBG. Experimental results are shown in red and the design in black.
Fig. 9
Fig. 9 Amplitude of the excess a) transmission and b) reflection loss of a square-shaped dispersion-less MZI-IBG. The experimental results are in red and the optimal fit (Δκ = 200 m−1, SRA1 = 8% and SRA2 = 4% and Δneff = 1x10−4) is shown in black.

Equations (8)

Equations on this page are rendered with MathJax. Learn more.

[ E out,1 E out,2 ]=[ A B C D ][ E in,1 E in,2 ]
[ E z 0 + E z 0 ]=[ 1 t r * t * r t 1 t * ][ E z 0 + L g + E z 0 + L g ],
E 1 =( A 1 A 1 r 1 + B 1 C 1 e i2Δ ϕ in r 2 ) e i2 ϕ in E 2 =( A 1 C 1 r 1 + C 1 D 1 e i2Δ ϕ in r 2 ) e i2 ϕ in E 3 =( A 1 A 2 t 1 + C 1 B 2 e iΔ ϕ in e iΔ ϕ out t 2 ) e i ϕ in e i ϕ out E 4 =( A 1 C 2 t 1 + C 1 D 2 e iΔ ϕ in e iΔ ϕ out t 2 ) e i ϕ in e i ϕ out
E 1 = 1 2 ( r 1 r 2 ) e i2 ϕ in E 3 = 1 2 ( t 1 t 2 ) e i ϕ in e i ϕ out E 2 = 1 2 ( r 1 + r 2 )i e i2 ϕ in E 4 = 1 2 ( t 1 + t 2 )i e i ϕ in e i ϕ out
E 1 =r( A 1 A 1 + B 1 C 1 ) e i2 ϕ in E 3 =t( A 1 A 2 + C 1 B 2 ) e i ϕ in e i ϕ out E 2 =r( A 1 C 1 + C 1 D 1 ) e i2 ϕ in E 4 =t( A 1 C 2 + C 1 D 2 ) e i ϕ in e i ϕ out .
ABCD=[ 1K i K i K 1K ]
E 1 =r( 12 K 1 ) e i2 ϕ in E 2 =2r K 1 1 K 1 i e i2 ϕ in E 3 =t( 1 K 1 1 K 2 K 1 K 2 ) e i ϕ in e i ϕ out . E 4 =t( 1 K 1 K 2 + K 1 1 K 2 )i e i ϕ in e i ϕ out
E 1 = r 2 ( 1 e i2Δ ϕ in ) e i2 ϕ in irΔ ϕ in e i2 ϕ in E 2 = r 2 ( 1+ e i2Δ ϕ in )i e i2 ϕ in r( 1+iΔ ϕ in )i e i2 ϕ in E 3 = t 2 ( 1 e iΔ ϕ in e iΔ ϕ out ) e i ϕ in e i ϕ out t Δ ϕ in +Δ ϕ out 2 i e i ϕ in e i ϕ out , E 4 = t 2 ( 1+ e iΔ ϕ in e iΔ ϕ out )i e i ϕ in e i ϕ out t( 1+i( Δ ϕ in +Δ ϕ out 2 ) )i e i ϕ in e i ϕ out

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