## Abstract

We demonstrate, in both theory and experiment, 4-port, electrically tunable photonic filters using silicon contra-directional couplers (contra-DCs) with uniform and phase-shifted waveguide Bragg gratings. Numerical analysis, including both intra- and inter-waveguide couplings, is performed using coupled-mode theory and the transfer-matrix method. The contra-DC devices were fabricated by a CMOS-photonics manufacturing foundry and are electrically tunable using free-carrier injection. A 4-port, grating-based photonic resonator has been obtained using the phase-shifted contra-DC, showing a resonant peak with a 3-dB bandwidth of 0.2 nm and an extinction ratio of 24 dB. These contra-DC devices enable on-chip integration of Bragg-grating-defined functions without using circulators and have great potential for applications such as wavelength-division multiplexing networks and optical signal processing.

© 2013 Optical Society of America

## 1. Introduction

Bragg gratings are fundamental components for a wide range of applications in optical communications and sensing systems [1], such as wavelength-division multiplexing (WDM) filters [2], dispersion engineering [3], lasers [4], optical signal processing [5], and optical sensors [6]. In the last decade, there have been significant efforts put into the development of waveguide Bragg gratings (WBGs) for the silicon platform [7–9] due to its great potential for large-scale photonic and electronic integration using established complementary metal-oxide-semiconductor (CMOS) facilities. These WBG devices include bandpass filters [2, 10] and high-*Q* transmission filters [11, 12].

However, most demonstrated WBG devices operate in reflection mode (i.e., as 2-port devices). This brings about the challenging requirement to integrate an optical circulator, which not only increases the cost and complexity of the integrated photonic circuits but also is difficult to integrate on the silicon platform. Grating-assisted asymmetric waveguide couplers have no, or very weak, reflection at the operating wavelength, therefore, they intrinsically function as wavelength-selective add-drop filters (4-port devices), circumventing the need for optical isolators or circulators [13]. Particularly, asymmetric couplers with contra-directional coupling between two waveguides, namely, contra-directional couplers (contra-DCs), are very attractive due to their compactness. The grating-assisted contra-DCs in planar optical waveguides were first investigated based on the silica and III–V materials [13–17] and have recently been implemented on the silicon-on-insulator (SOI) platform, either in sidewall-modulated strip waveguides [18] or slab-modulated rib waveguides [19]. SOI waveguides with a typical height of 200 to 300 nm have much higher dispersions as compared to conventional optical waveguides, such as optical fibres and planar silica waveguides, therefore, one can obtain large effective-index differences between two SOI waveguides by simply varying their widths. Using such asymmetric waveguides, we can obtain contra-DC-based filters with better performance such as higher out-of-band rejection ratio and wider usable spectral range [20]. Integrated silicon photonic contra-DCs have enabled many novel applications such as on-chip optical-pulse compression [21], wide-bandwidth wavelength demultiplexers [22], single-longitudinal-mode microring resonators [23], single-band, flat-top add-drop filters [24], and 4-port photonic resonators [25]. These devices can be easily cascaded and integrated with other photonic components for large-scale integrated photonic circuits.

In this paper, we perform coupled-mode analysis on contra-DCs with uniform and phase-shifted Bragg gratings. The models used in previous work on silicon photonic contra-DCs [18, 19] typically neglected the intra-waveguide reflections; this assumption is based on the arguments that Bragg wavelengths are generally far (> 10 nm) from the inter-waveguide coupling wavelength (drop-port central wavelength) and that the waveguides are not strongly corrugated. In this paper, we use a more comprehensive model [15, 16], taking the intra-waveguide back reflections, as well as the mode transitions between the individual waveguides and the coupler region, into consideration. This allows us to reveal the behaviour of the contra-DC in a large spectral range, not limited to the relatively small range near the drop-port central wavelength, as well as to see the effects of the intra-waveguide back reflections on the drop-port response. We further experimentally demonstrate an electrically tunable, 4-port photonic filter using a phase-shifted contra-DC fabricated by a CMOS-photonics foundry and compare the experimental results with theory. We also demonstrate the electrical properties, such as wavelength tuning efficiency and modulation response, of the fabricated device and compare them with numerical models obtained using a semiconductor optoelectronic simulation tool [26].

This paper is organized as follows: in Section 2, we briefly review the principle of add-drop filters using contra-DCs and discuss important aspects in the design of contra-DC devices; in Section 3, we present the coupled-mode models describing contra-DCs with uniform and phase-shifted Bragg gratings; in Section 4, we describe our designed device structures and parameters; Section 5 shows the numerical simulation using an eigenmode solver [27] and the coupled-mode equations described in Section 3, as well as the modeling of the electrical performance; in Section 6, we demonstrate the experiment and compare the measured results with theory; in Section 7, we discuss about potential applications of the contra-DCs; Section 8 is the conclusion; more details used in the simulation are given in the Appendix.

## 2. Principle of Add-Drop Filters Using Contra-DCs

A contra-DC-based add-drop filter is illustrated in Fig. 1. The device consists of two dissimilar optical waveguides which form an asymmetric coupler having a periodic perturbation or Bragg grating formed either on the waveguides (e.g., by corrugating the sidewalls or the top surfaces) or between them (e.g., by corrugating the cladding or the slab in the coupler gap). According to the transverse eigenmodes, the filter can be divided into the coupler region (Region II) and the mode-transition regions (Regions I and III) which play different roles in determining the filter performance:

**Coupler region**(Region II) is the essential wavelength-selective component in which the grating pitch is chosen so that a specific wavelength,*λ*, is dropped due to the grating-assisted, contra-directional coupling between the forward-propagating, lowest-order transverse mode ( ${\mathbf{E}}_{\mathbf{1}}^{+}$) and the backward-propagating, next-higher-order transverse mode ( ${\mathbf{E}}_{\mathbf{2}}^{-}$) of the coupler, following the phase-match condition [13, 28]:_{D}Coupling also exists between the forward-propagating and backward-propagating waves for each mode, resulting in intra-waveguide back reflections centered at the Bragg wavelengths (

In order to suppress the effect of the intra-waveguide back reflections on the dropped signal, the central wavelengths (*λ*_{r1}for**E**and_{1}*λ*_{r2}for**E**) determined by the Bragg condition [13, 28]:_{2}*λ*,_{D}*λ*_{r1}, and*λ*_{r2}) should be widely separated, therefore, the coupler is typically designed to be highly asymmetric. Varying the widths of high-index-contrast SOI waveguides, the spacing between*λ*and_{D}*λ*_{r1}(or*λ*_{r2}) can be extended to over 35 nm [20], covering the entire span of the conventional wavelength window, i.e., the C band, in fibre-optic communication. The coupled-mode equations and transfer functions of the coupler region are described in Section 3.**Mode-transition regions**(Regions I and III) are the regions for the transitions between the modes of the coupler (**E**and_{1}**E**) and those of individual waveguides (_{2}**E**and_{a}**E**). Ideal mode transitions result in_{b}**E**coupling only to_{a}**E**and_{1}**E**coupling only to_{b}**E**, i.e., no inter-waveguide coupling between_{2}**E**and_{a}**E**or between_{2}**E**and_{b}**E**should occur. However, even a small mode mismatch, between_{1}**E**and_{a}**E**or between_{1}**E**and_{b}**E**, may cause considerable co-directional coupling and, thus, stronger intra-waveguide back reflection. In order to obtain near ideal mode transitions, two important aspects need to be taken into consideration. Firstly, high coupler asymmetry is helpful to obtain a good match between_{2}**E**and_{a}**E**and between_{1}**E**and_{b}**E**. Secondly, well-designed tapers are desired for adiabatic mode transitions between the desired modes. The effect of the mode transitions will be further discussed in Section 5._{2}

## 3. Coupled-Mode Analysis

#### 3.1. Contra-DC with a Uniform Grating

As shown in Fig. 2, we consider the transverse modes, **E**_{1}(*x*, *y*) and **E**_{2}(*x*, *y*), of the unperturbed coupler, mainly confined in waveguide *a* (WG *a*) and waveguide *b* (WG *b*), respectively. The two modes can propagate in both the positive (+) and the negative (−) directions. Thus, the electric field in the contra-DC can be represented by

*β*̂ =

*β*−

*iα*represents the complex propagation constant;

*β*and

*α*are the real propagation constant and the propagation loss, respectively;

*A*

^{+},

*B*

^{+},

*A*

^{−}, and

*B*

^{−}are the field amplitudes as functions of the longitudinal position,

*z*. Assisted by the dielectric perturbation, contra-directional couplings may occur either between the two transverse modes (inter-waveguide coupling) or between a mode and the same mode that propagates in the opposite direction (back reflection). The grating pitch is chosen so that the co-directional coupling between the two transverse modes does not occur within the spectral range of interest. The coupled-mode equations are given by [15, 16]

*κ*

_{11}and

*κ*

_{22}are the coefficients for the back reflections of

**E**

_{1}and

**E**

_{2}, respectively;

*κ*

_{12}( ${\kappa}_{21}={\kappa}_{12}^{*}$) is the coefficient for the contra-directional coupling between

**E**

_{1}and

**E**

_{2}. These coupling coefficients are given by

*β*̂

_{1,2}=

*β*̂

_{1,2}−

*π*/Λ and Δ

*ε*

_{1}(

*x*,

*y*) is the first-order Fourier-expansion coefficient of the dielectric perturbation.

The coupled-mode equations can be solved by the transfer-matrix method, for which a schematic of the contra-DC is shown in Fig. 2. Given

*z*

_{0}and

*z*

_{1}is given by where

*C*is the transfer matrix of the contra-DC with a uniform grating, for which an analytical solution can be obtained by solving Eqs. (4a)–(4d) and is given by [14, 16]

*S*

_{1}and

*S*

_{2}are given by Eqs. (9) and (10), respectively.

#### 3.2. Phase-Shifted Contra-DC

The schematic of a phase-shifted contra-DC is shown in Fig. 3. The structure may be viewed as a contra-DC with a defect (i.e., an additional quarter-wave layer) at the centre of the periodic perturbation. As we will see in the next section, the presence of this defect introduces a *π* shift in the phase response of the drop-port. This defect or phase shift creates a ring-like optical cavity with distributed optical feedback provided by the grating-assisted, contra-directional coupling; as a result, the structure acts as a photonic resonator with the resonant peak located at the centre of the stop-band of the drop-port response. Different from conventional phase-shifted Bragg gratings (2-port devices), where the optical resonance happens inside a single waveguide, this contra-DC photonic resonator has a circular feedback loop between two waveguides, as highlighted in Fig. 3, enabling a narrowband add-drop filter (4-port device).

From the perspective of the transfer-matrix analysis, the structure can also be viewed as a half-wave (Λ) layer sandwiched by two contra-DCs; accordingly, the matrix equation relating the fields at the two ends of the structure is given by

*P*(

*z*

_{2},

*z*

_{3}) is the propagation matrix of the phase shift and is given by

The round-trip phase of the cavity is

where*δ*

_{c1}and

*δ*

_{c2}are the phases of the two contra-DCs. Then we can see the device is resonant at

*λ*since the phase-match condition, given by Eq. (1), ensures that

_{D}Critical coupling of a ring cavity is defined as the condition at which the through-port transmission reaches zero [30]. Similarly, the critical coupling condition of the phase-shifted contra-DC is given by

where A*is the round-trip power attenuation. For the symmetric cavity design, i.e.,*

_{rt}*η*

_{c1}=

*η*

_{c2}, critical coupling can never be obtained due to optical losses, although a high extinction ratio can be easily obtained as long as

*η*

_{c1,c2}<< A

*.*

_{rt}## 4. Device Design

Figure 4 shows the schematic of the cross-section of the contra-DC. Devices with both uniform and phase-shifted gratings are designed based on an SOI rib-waveguide geometry with a rib height of 210 nm and a slab thickness of 110 nm. Each coupler has an input/through waveguide width, *W _{a}*, of 600 nm, an add/drop waveguide width,

*W*, of 400 nm, and a coupler gap,

_{b}*G*, of 200 nm. The dielectric perturbation is formed on both the sidewalls of the waveguide ribs and the slab between them to achieve a strong coupling. The corrugation amplitudes on the rib sidewalls, Δ

*W*and Δ

_{a}*W*, are 50 nm and 30 nm, respectively. The grating pitch, Λ, is 300 nm with a duty cycle of 50%. The period number,

_{b}*N*, is chosen to be 700, corresponding to a device length of 210

*μm*. For the phase-shifted contra-DC, an additional 1/4-

*λ*high-index layer is introduced at the centre of the coupler (i.e., there are 350 grating periods on each side of the phase-shift). Linear tapers are used between the rib waveguides and the routing waveguides (210-nm-high strip waveguides). A p-i-n configuration is used for frequency tuning. The n+ (5.5 × 10

^{18}cm

^{−3}) and p+ (5.7 × 10

^{18}cm

^{−3}) regions are 200 nm away from the edges of the waveguide ridges. Thus, the intrinsic region includes 600 nm of slab and 1

*μ*m of rib waveguide, for a total distance of 1.6

*μ*m.

## 5. Simulation

#### 5.1. Mode Analysis

The first and second transverse modes of the contra-DC were calculated using an eigenmode solver [27] and are shown in Fig. 5. The central wavelengths of the contra-directional coupling and the intra-waveguide reflections can be found following the phase-match conditions given by Eqs. (1) and (2):

The coupling coefficients were calculated using Eqs. (5a)–(5c), assuming a sinusoidal distribution of the dielectric perturbation along the longitudinal direction, and are listed in Table 1. The predicted back reflections are stronger than the inter-waveguide contra-directional coupling since the modes have stronger overlaps with themselves as compared to each other. A sinusoidal distribution is used due to lithography smoothing [31].In addition to the intra-waveguide reflective couplings (*κ*_{11} and *κ*_{22}), the mode transitions or couplings between the individual waveguides and the coupler region also have considerable impact on the magnitudes of the intra-waveguide back reflections, as we will see in the simulation results shown below. If no taper is used in the mode-transition regions, the coupling coefficients between the modes of the individual waveguides (**E*** _{a}* and

**E**

*) and that of the coupler region (*

_{b}**E**and

_{1}**E**) can be estimated using the mode overlap calculation [27]:

_{2}#### 5.2. Spectral Responses

### 5.2.1. Contra-DC with a uniform grating

Figures 6(a)–6(d) shows the calculated spectra of the contra-DC with a uniform grating obtained by solving the differential equations Eqs. (4a)–(4d) (the transfer-matrix solutions are given in the Appendix), with and without considering the intra-waveguide reflections and mode transitions.

In Fig. 6(a), we can see that, when no intra-waveguide back reflection is considered, the through-port and drop-port responses are analogous to the transmission and reflection of a 2-port uniform Bragg grating. Figure 6(b) shows the drop-port phase response, *ϕ*, as well as the group delay, *τ*, calculated by [28]:

*ω*is the optical frequency. Again, the response of the drop port is analogous to that of the reflection of a uniform Bragg grating.

Figure 6(c) shows the through-port and drop-port responses of the contra-DC when intra-waveguide reflections are considered and ideal mode transitions between the individual waveguides and the coupler region are assumed. A deep notch appears in the through-port spectrum at *λ*_{r1}, in good agreement with the value calculated using the phase-match condition given by Eq. (17b); accordingly, strong reflection at the input port is seen in Fig. 6(d). The add-port response due to the intra-waveguide reflection is also shown in Fig. 6(d), having strop-band features at *λ*_{r1} and *λ*_{r2}.

Figures 6(e) and 6(f) show the spectra when both the intra-waveguide reflections and the mode transitions are included in the simulation. The coupling coefficients of the mode transitions are calculated using Eq. (18); this is the worst case assuming no taper is used in the mode-transition region. The coupling coefficients calculated are listed in Table 1. Ideally, there should be no coupling between **E _{a}** and

**E**or between

_{2}**E**and

_{b}**E**. However, the calculated value of

_{1}*k*

_{b}_{1}is very high, corresponding to a power coupling of over −12 dB between

**E**and

_{b}**E**. The effect of this high undesired coupling can be seen in the drop-port response: the power response at

_{1}*λ*

_{r2}is increased from below −30 dB (Fig. 6(c)) to over −20 dB (Fig. 6(e)); in addition, the response is slightly distorted and the input reflection is increased at

*λ*due to the mode transitions between the modes of the coupler region and that of the individual waveguides. The mode-transition-related deterioration can be avoided by a well-designed tapering structure where co-directional couplings are significantly suppressed.

_{D}### 5.2.2. Phase-shifted contra-DC

The spectral responses of the phase-shifted contra-DC were calculated using Eqs. (4a)–(4d), (12), and (13) (the transfer-matrix solutions are given in the Appendix) and are shown in Figs. 7(a)–7(f). Due to the photonic cavity created by the phase shift, a sharp resonant peak appears at the centre of the stopband of the through-port response and, accordingly, a deep notch appears in the drop-port response. In the ideal case, when there is no intra-waveguide reflection or optical loss, the through port has a zero in the power transmission at the resonant wavelength (*λ _{D}*), corresponding to the critical coupling condition given by Eq. (16); accordingly, the phase response has a sharp

*π*shift across

*λ*, as shown in Fig. 7(b), which can be used for optical temporal differentiation [5]. The effect of optical loss (5 dB/cm was used in the simulation) can be seen in Figs. 7(c) and 7(d); due to the deviation from the critical coupling condition, the notch depth is reduced from infinity (in dB) to about 24 dB and the phase shift at the resonance is smoothed. The responses are further distorted when the intra-waveguide reflections are considered, as shown in Figs. 7(e) and 7(f); the strong back reflections cause the frequency shift and splitting of the resonant peak, as well as the deterioration of the phase response (the phase-shift magnitude is significantly suppressed and the transition edge is blurred).

_{D}## 6. Experiment

The devices, with the design described in Section 4, were fabricated by BAE Systems via the OpSIS foundry service [32]. Fibre grating couplers [33] were used for the optical measurement.

#### 6.1. Spectral Responses

The measured through-port and drop-port spectra of a contra-DC with a uniform grating are shown in Fig. 8. The spectra were normalized using the transmission spectrum of a pair of fibre grating couplers. The phase-match wavelengths *λ _{D}*,

*λ*

_{r1}, and

*λ*

_{r2}are about 1519 nm, 1548 nm, and 1494 nm, respectively, slightly different from the predicted values due to the wafer nonuniformity, as shown in Fig. 5. The drop-port response shows a flat-top passband at

*λ*with a 1-dB bandwidth of about 3.4 nm. The bandwidth between the first two notches beside

_{D}*λ*is about 3.8 nm, smaller than the predicted value of 4.4 nm, which indicates a smaller coupling coefficient due to the smoothing effects of the lithography and fabrication [2, 19, 31]; the distortions arising from the fabrication process can be accounted for in the lithography simulation to obtain good agreement between theory and experiment [31]. The through-port spectrum shows a deep notch at

_{D}*λ*with an extinction ratio of over 20 dB, limited by the noise floor of the measurement. The notch in the through-port at

_{D}*λ*

_{r1}is about 7 dB, significantly lower than the simulation shown in Fig. 6(e). The back reflection at

*λ*

_{r2}shown in the drop-port spectrum is also weaker than the simulation result. To fit the measured results,

*κ*

_{11}and

*κ*

_{22}have been reduced by over 10 times the predicted values, while

*κ*

_{12}is in good agreement with the calculation (listed in Table 1); this is also different from our previous experimental results [19, 20]. One possible reason for the unexpected low intra-waveguide reflections is the misalignment between the perturbation on the sidewalls and that on the slab, which was unintentionally introduced during the fabrication since the 210-nm (rib) and 110-nm (slab) silicon layers were fabricated using multiple lithographic and etch steps. This could create an antireflection effect due to destructive interference [24]. Also, the use of linear tapers is helpful to improve the mode transitions between the coupler region and individual waveguides and, therefore, reduces these reflections. This results is fortunate, since low back reflections are favourable to reduce the distortions of the contra-directional coupling at

*λ*. Future designs could incorporate this antireflection approach to further improve the contra-DC performance [24]. To fit the measured results, we have adjusted the parameters (listed in Table 1) in the simulation and obtained excellent agreement with experiment, as shown in Fig. 8.

_{D}The measured and fit spectra of a phase-shifted device are shown in Fig. 9. A resonant peak is clearly seen at the centre (1519.6 nm) of the stopband of the through-port spectrum and a corresponding notch is in the drop-port spectrum. The transmission peak in the through-port has an out-of-band rejection ratio of more than 17 dB and a 3-dB bandwidth of 0.2 nm, corresponding to a quality factor (*Q*) of about 7,000; in contrast to the simulated results shown in Fig. 7(c), no resonance splitting or distortion is observed, also indicating that the intra-waveguide back reflections are weak. The resonant notch in the drop-port response has an extinction ratio of over 24 dB. These performance metrics can further be improved by optimizing the coupling strength and the coupler length. Using the transfer-matrix method given in Section 3.2, with the same parameters as above, listed in Table 1, an excellent fit to the measured results has been achieved, as shown in Fig. 9. It is worth emphasizing that this photonic filter is resonant at the central wavelength of the drop-port response (*λ _{D}*), i.e., at the phase-match condition of the contra-directional coupling and is significantly detuned (by over 20 nm in this case) from the Bragg condition of the intra-waveguide reflection at

*λ*

_{r1}; this means that, as opposed to conventional transmission filters using Bragg cavities (e.g., vertical-cavity lasers or Bragg waveguides [11]), no, or very weak back reflections occur at the operating wavelength or within the stopband.

#### 6.2. Electrical Tuning

The phase-shifted contra-DC is electrically probed and the spectra for various tuning currents are shown in Fig. 10 (a). As the current increases, the spectrum shows a blue shift due to the plasma dispersion effect [34]. The device yields a wavelength shift of 0.8 nm for a current of 1.5 mA and operates under 1.0 V. This tuning response is plotted in Fig. 10 (b) and has a tuning coefficient of −0.73 nm/mA. The current is uniformly injected along the entire length of the coupler region, therefore, the electrical tuning shifts both the Bragg stopband (the stopband edges are not shown in the figure) and the resonant mode, simultaneously. For large tuning currents, free-carrier absorption introduces excess loss and thus, reduces *Q* and the extinction ratio. For example, at 1.5 mA, *Q* is reduced to 2,000. At 0.08 mA, the spectrum shows a higher extinction ratio as compared to the zero-bias condition, which is unexpected according to coupled-resonator theory, as discussed in Section 3.2, since the phase-shift is designed to be right in the middle of the contra-DC and injecting current is supposed to reduce the extinction ratio. This unexpected higher extinction ratio at 0.08 mA is mostly likely due to fabrication errors that caused a slight asymmetry between the coupling coefficients, i.e., *η*_{c1} < *η*_{c2}. As a result, the coupling condition at 0.08 mA is closer to the critical coupling condition, given by Eq. (16), as compared to the zero bias. The tuning efficiency and I–V curve were calculated using a finite-element software package [26] and are shown in Figs. 10(b) and (c). The small-signal modulation frequency response of the device is shown in Fig. 10 (d) and has a 3-dB bandwidth of 90 MHz limited by the lifetime of the injected carriers.

## 7. Discussion

#### 7.1. Optimization

The uniform-grating contra-DCs have flat-top responses, however, their sidelobes are very high and may cause significant crosstalk if they are used in a WDM system. Apodization is a well-know technique to tailor the spectra of Bragg gratings and suppress the sidelobes. For example, Fig. 11 shows the simulated spectra of an apodized contra-DC using a Gaussian-shape grating profile with the coupling coefficient of the *n*th period given by

*κ*= 16.3 × 10

_{max}^{3}m

^{−1},

*a*= 17, and

*N*= 1, 200 have been used in the simulation (other parameters are the same as those listed in Table 1); we can see that a near-ideal filtering profile can be obtained with sidelobes below −40 dB.

Another issue that may weaken the performance of contra-DCs as ideal add-drop filters comes from the intra-waveguide back reflections. The spacing between *λ _{D}* and

*λ*

_{r1}(or

*λ*

_{r2}) can not be increased indefinitely by only increasing the coupler asymmetry, without the tradeoff of higher optical losses and possible high-order modes. There are two methods to solve this reflection issue. Firstly, the tapers used in the mode-transition regions should be carefully designed to suppress the undesired co-directional couplings. Secondly, the use of out-of-phase gratings on the external sides of the coupler can significantly suppress the back reflections [24].

For frequency tuning or modulation of the photonic resonator in the phase-shifted contra-DC, the speed or bandwidth is limited by the photon lifetime, carrier dynamics, and RC constant of the junction. The *Q* of the demonstrated photonic resonator is in a range of 2,000 to 7,000, indicating a photon-lifetime-limited bandwidth of over 25 GHz, therefore, its speed is mainly limited by the carrier lifetime of the p-i-n junction and can be significantly enhanced by using carrier depletion in a reversed-biased p-n junction [34]. In addition, the tuning efficiency or modulation efficiency can also be improved. In the case of the device demonstrated in this paper, the current is uniformly injected into the entire length of the coupler region, whereas an optimal design is anticipated to have the maximal overlap of the current density with the longitudinal optical intensity distribution, for which a localized or modulated doping profile may be desirable.

#### 7.2. Applications

Contra-DCs provide a promising solution to implement Bragg-grating defined functions in large-scale integrated photonic circuits for a variety of applications. For example, multiple contra-DCs with various grating pitches can be cascaded along a single bus waveguide to form a WDM multiplexer or demultiplexer. Compared to other add-drop filters, e.g., using cascaded microring resonators, the contra-DCs can have much wider bandwidths (> 10 nm) with lower insertion losses [24]. This is attractive because a wider channel bandwidth allows higher tolerance to the wavelength drift, due to temperature fluctuations and wafer non-uniformity, especially considering that high wavelength sensitivity to temperature and wafer non-uniformity may cause significant reliability issues and has been a large obstacle to improving the power budget of WDM networks using CMOS-photonics technologies [8, 35]. Apodized Bragg gratings have recently been demonstrated to obtain electrically tunable delay lines [36]. Contra-DCs can also be used for optical buffering applications and, importantly, can be easily cascaded and integrated with other photonic components since they do not introduce strong reflection at the operating wavelength.

In addition to modulators and tunable filters, phase-shifted contra-DCs can be used for many applications conventionally based on reflection operation of phase-shifted Bragg gratings, such as temporal differentiation [5], laser linewidth reduction [37], and microwave generation and signal processing [38]. Replacing two-port Bragg gratings with contra-DCs allows direct integration (i.e., without using circulators) with other photonic devices such as lasers and modulators, as well as the combining of these functions to realize on-chip optical signal processors and photonic microwave systems. Beyond the photonic resonator with a single phase-shift, as demonstrated in this paper, multiple phase shifts can be inserted into a contra-DC to obtain a series of resonant peaks for multi-wavelength operation. As compared to a microring resonator, where all the resonant peaks (longitudinal modes) are determined by the same cavity, the resonant peaks of contra-DCs can be controlled individually by adjusting each phase shift separately, enabling full design control of mode spacings and mode positions.

## 8. Conclusion

In summary, we have demonstrated a systematic analysis of a class of Bragg-grating based, 4-port photonic filters, namely silicon photonic contra-DCs, using coupled-mode theory and the transfer-matrix method. We have investigated important aspects, such as intra-waveguide back reflections and mode-transitions, predicting the device performance. We have also experimentally demonstrated electrically-tunable contra-DCs with both uniform and phase-shifted gratings using CMOS-photonics fabrication technology. These devices are very compact and integration-friendly, therefore, they have great potential for integrating Bragg-grating-defined functions into large-scale integrated photonic circuits for a wide range of applications such as WDM optical interconnects, optical signal processing, and microwave photonics.

## Appendix

In this section, we derive the transfer-matrix solutions for the coupled-mode equations given by Eqs. (4a)–(4d), i.e., how to solve Eqs. (7) and (12). To facilitate the calculation, we rewrite *E*(*z*):

Then, for a contra-DC device which has one uniform grating or multiple sections (gratings or phase-shifts), the matrix solution is given by [14, 16]

*M*is the total transfer matrix and

*M*

^{++},

*M*

^{+−},

*M*

^{−+}, and

*M*

^{−−}are 2 by 2 submatrices of

*M*. Equation (23) is the general form of Eqs. (7) and (12), i.e., for the uniform-grating contra-DC,

*M*=

*C*, while for the phase-shifted contra-DC,

*M*=

*C*×

*P*×

*C*. With a transformation of Eq. (23), the relationship between the input fields and the output fields is given by

*E*from the only input, then the fields incident to the coupler region are given by

_{in}*E*,

_{Thru}*E*,

_{Add}*E*, and

_{R}*E*, corresponding to the through-port, addport, input-reflection, and drop-port responses, respectively) of the contra-DC-based add-drop filter are given by

_{Drop}## Acknowledgments

We acknowledge BAE Systems for the fabrication; Yi Zhang and Ran Ding for their help on the measurement; Lumerical Solutions, Inc., for the modelling software, and Dylan McGuire, for assistance with device modelling; CMC Microsystems for the software resources, particularly for Mentor Graphics Pyxis and Calibre; the Natural Sciences and Engineering Research Council of Canada for their financial support. We would further like to thank Gernot Pomrenke of the Air Force Office of Scientific Research, Juan Rey, Chris Cone, John Ferguson and their colleagues at Mentor Graphics, as well as Mario J. Paniccia of Intel Corporation for their ongoing support of the OpSIS program and of this work.

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