1. Introduction

In a recent paper [1], we have shown the interest in the use of Bragg grating (BG) dispersion for achieving co-directional coupling between mismatched waveguides. In this mode of operation, the BG is operated near band gap, and co-directional phase matching is obtained for grating coupling coefficients higher than a certain threshold value. The analysis in [1] has also shown that the use of the BG structural dispersion could improve the wavelength selectivity of an asymmetric directional coupler beyond the limit imposed by the waveguide material dispersion. A selectivity enhancement by a factor of four has been predicted as compared to state-of-the-art results in the conventional approach of highly asymmetric waveguides [2–4].

However, previous analysis [1] was carried out for the case where the differential dispersion between the two waveguides was low. This typically corresponds to a horizontal type coupling geometry [5–7], which is not the best one in terms of high wavelength selectivity. A natural question then arises: what could be expected from the additional use of a BG structural dispersion in a coupler structure with high differential dispersion between the guides? Such a structure typically corresponds to a vertical type coupling geometry (Fig. 1 ), where the core layers of the two guides are made of different materials [2–4]. This is indeed with this coupling geometry that state-of-the-art coupler performances were reported. Because of the high differential dispersion between the guides, co-directional phase matching can be practically attained in the absence of any BG. The aim of the present work is then to answer the question formulated above and investigate the additional possibilities offered by the control of the structural dispersion in a system of two highly asymmetric coupled waveguides with high differential dispersion.

 

Fig. 1 Sketch of the vertical coupling geometry Bragg grating assisted asymmetric directional coupler (BGAADC).

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Intuitively, one may think that the use of an additional dispersion simply translates into a better coupler selectivity through an increase of the waveguide group index. However, we will show that the situation is much more complex than in the case of a low differential dispersion between the guides. In most cases, a minimum BG coupling strength is required for the enhancement of the wavelength selectivity. This threshold condition is surprisingly similar to that encountered in the case of initially mismatched waveguides [1]. In contrast, for coupling strengths larger than threshold, the presence of the BGs leads to the appearance of two transmission bands instead of one. The existence of these two solutions for co-directional phase matching is explained by the combination of an initial phase matching due to the high asymmetry of the bare waveguides and the opening of a photonic band gap due to the Bragg grating periodicity. This indeed leads to a much stronger curvature of the dispersion characteristics than in the case of initially mismatched waveguides. A major result for practical applications is that the BG induced transmission band can be at least 5.5 times narrower than the original transmission band in the absence of BG.

In the particular case where the Bragg frequency is close to the phase matching frequency in the absence of BG, it will be shown that the dual band operation can be achieved without any threshold condition on the BG coupling coefficient. The two transmission bands are then quasi-identical, each bandwidth being two times narrower than that obtained in the absence of BG. This symmetrical dual band operation, which is independent of the BG coupling strength, is expected to offer other functionalities for photonic integrated circuits.

The paper is organized as follows. The analysis of phase-matching conditions for co-directional coupling near the BG band gap is presented in Section 2. Results from the analytical model are verified in section 3 using a coupled mode theory (CMT) four-wave model. The selectivity performances of the dual transmission band coupler are illustrated in sections 4 and 5 for the cases with and without threshold condition, respectively. This is followed by a summary and concluding remarks in section 6.

2. Bragg grating assisted co-directional phase matching

In the vertical configuration chosen to illustrate a high differential dispersion between waveguides (Fig. 1), the asymmetric directional coupler (ADC) consists of two superimposed waveguides with Bragg gratings, which can be arbitrarily represented as periodic side edge corrugations. In fact, any type of BG implementation such as side- or top- etched corrugations, photo-imprinted refractive index modulation, etc. can be indifferently envisaged. Unlike many optical add-drop multiplexers (OADM) based on two coupled waveguides with BG assisted coupling [8–19], the device under study is assumed to be operated outside the BG band gap while its selectivity is expected to benefit from the dispersive properties of the BGs.

The starting point of our analysis is the examination of the phase-matching conditions in a system of two coupled waveguides assisted by BGs. For this purpose, we use an effective medium approach [20, 21] where the BG is treated as an uniform homogeneous medium with dispersion properties identical to those of the real grating. Assuming a periodic corrugation with a cosine profile for the sake of simplicity, the relation given by Yariv et al. [22] shows that the equivalent propagation constant of a waveguide is a function of the wave-vector k=2π/λ, of the grating period Λ, of the BG coupling coefficient χ, and of the waveguide effective index n_eff:

The refractive index itself is a function of the wave-vector. By definition, phase matching means that for a given wave-vector, propagation constants in both waveguides are equal, i.e. β _{Eq. (1)}= β _{Eq. (2)}. From Eq. (1), it follows that:

In the most general case, Eq. (2) cannot be solved analytically. Yet, Eq. (2) can be greatly simplified by assuming that both gratings have the same period Λ₁=Λ₂=Λ and that for both waveguides, dispersion relations can be locally approximated by linear functions. Following this (see Appendix A), wave-vector solutions of Eq. (2) express as:

2.1 Bragg grating on the waveguide with higher group index (n_g1>n_g2)

Figures 2(a) and 2(b) show the waveguide dispersion characteristics, ω = ω(β_eq), calculated for n_g1>n _g2. In Fig. 2(a), the grating period Λ is chosen in such a way that the Bragg frequency ω_Br1 is larger than ω_ADC, which is the phase-matching frequency in the absence of the BG. The reverse situation (ω_ADC>ω_Br1) is shown in Fig. 2(b). In both figures, the black continuous curve is the dispersion characteristic of the uniform waveguide without grating (χ ₂=0). The family of hyperbola-like curves represents the BG waveguide dispersion curves for different values of χ ₁. The branches drawn in solid lines correspond to forward propagating waves. Those in dashed lines correspond to backward propagating waves shifted in abscissa by a grating vector Q=2π/Λ. Colors from blue to light green in the graph correspond to increasing values of χ ₁.

 

Fig. 2 Dispersion curves of the uniform waveguide (black) and the BG assisted waveguide for increasing values of the coupling coefficient χ₁ (from blue to light green) when n_g1>n_g2. (a) ω_ADC<ω_Br1 (b) ω_ADC>ω_Br1. Insets show enlarged views in the vicinity of the phase matching wavelengths.

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Inspection of Fig. 2(a) reveals the presence of two intersections between the dispersion curves of the two waveguides near the Bragg frequency. One intersection, located below ω_Br1, corresponds to the phase matching condition initially obtained in the absence of the BG (χ ₁=0). Its position moves toward lower and lower frequencies as χ ₁ is increased. The second intersection, located above ω_Br1, is directly associated to the presence of the BG. Its position moves toward higher and higher frequencies as χ ₁ is increased. For small values of the coupling coefficient χ ₁, this second intersection point remains located in the region of backward propagating waves (k<π/Λ). This gives evidence of a contra-directional phase matching via the Bragg exchange coupling mechanism.

As the Bragg coupling coefficient increases, the curvature of the dispersion characteristic of the BG waveguide also increases in the vicinity of the BG band gap. For a sufficiently high value of the coupling coefficient, the dispersion curve of the uniform waveguide now intersects the BG waveguide dispersion curve in the region of forward propagating waves above the BG band gap (k>π/Λ). The red bold line in Fig. 2(a) corresponds to the threshold value ${\chi }_1={\chi }_{th}^{\prime }$ for co-directional phase matching. As is seen, the co-directional phase-matching condition is fulfilled for any value of the coupling coefficient larger than ${\chi }_{th}^{\prime }$.

For the situation depicted in Fig. 2(b) (ω_ADC>ω_Br1), phase matching due to the BG occurs at a frequency below ω_Br1. Except for that, the behavior is similar to that previously encountered for ω_ADC<ω_Br1. A minimum coupling coefficient ${\chi }_{th}^{\prime }$ is required for the existence of two phase matching frequencies.

2.2 Bragg grating on the waveguide with lower group index (ng1<ng2)

Figures 3(a) and 3(b) show the waveguide dispersion curves when the BG is placed on the waveguide with lower group index (n_g1<n_g2). Figure 3(a) corresponds to the situation where ω_ADC<ω_Br1. In this situation, dispersion curves of the two guides only intersect at frequencies below ω_Br1. When the coupling coefficient χ ₁ is small, i.e. when the curvature of the BG waveguide dispersion curve is weak, the dispersion curve of the uniform waveguide intersects that of the BG waveguide one time in the region of co-directional coupling (k<π/Λ) and a second time in the region of contra-directional coupling (k>π/Λ).

 

Fig. 3 Dispersion curves of the uniform waveguide (black) and the BG assisted waveguide for increasing values of the coupling coefficient χ₁ (from blue to light green) when n_g1<n_g2. (a) ω_ADC<ω_Br1 (b) ω_ADC>ω_Br1. Insets show enlarged views in the vicinity of the phase matching wavelengths.

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At a certain value of the coupling coefficient ${\chi }_1={\chi }_{th}^{″}$, i.e. for a certain curvature of the BG waveguide dispersion curve (red curve in Fig. 3a), the waveguide dispersion curves intersect two times in the region of co-directional phase matching (see insert). In other words, there are two frequencies at which co-directional phase matching occurs. An important feature is that the co-directional phase-matching condition is fulfilled only within a limited range of χ ₁ values: ${\chi }_{th}^{″}<{\chi }_1<{\chi }_d$. The upper limit χ _d corresponds to a tangential contact between the dispersion characteristics of the two guides. Therefore, for χ ₁>χ _d, co-directional phase matching can no longer be obtained. It is worthwhile mentioning that contra-directional phase matching is also excluded in this case.

When ω_ADC>ω_Br1 (Fig. 3(b)), the behavior is similar to that previously described for ω_ADC<ω_Br1 except for the fact that phase matching conditions are now achieved at frequencies above the BG band gap.

2.3 Analytical expressions of phase matching conditions

Expressions of the threshold values of χ ₁ for co-directional phase matching (${\chi }_{th}^{\prime }$and${\chi }_{th}^{″}$) can be derived from Eq. (2) by locally approximating the dispersion relations of the guides with linear functions and assuming that: χ ₁≥χ ₂>0. Details on formula derivation can be found in Appendix B. One important result is that a unique threshold formula can be used for ${\chi }_{th}^{\prime }$and${\chi }_{th}^{″}$, i.e. whether n _g1>n _g2 or n _g1<n _g2:

Equation (4) shows that the presence of a second BG (χ ₂≠0) increases the threshold value for co-directional phase matching. Such a situation with stronger BG index modulation is less favorable in practice. Co-directional phase matching becomes even impossible when the two BGs are of equal strength in our asymmetric coupler geometry. Therefore, one can conclude that co-directional coupling is optimally achieved with the use of only one BG in one of the two ADC waveguides. The hypothesis χ ₂=0 will then be kept for the rest of the paper.

Equation (4) can be rewritten in a form where the initial phase matching wavelength λ_ADC is explicitly included:

It then appears that the threshold value χ _th for co-directional phase matching increases with the separation between λ_ADC and the BG wavelength λ_{Br 1}. Conversely, χ _th decreases when λ_{Br 1} gets closer to λ_ADC, and even cancels when λ_{Br 1}=λ_ADC. The absence of threshold can also be directly viewed from Eq. (4). Indeed, the effective indexes of both waveguides are equal at the initial phase matching wavelength λ_ADC. The equality λ_{Br 1}=λ_ADC directly leads to λ_{Br 2}=λ_{Br 1}=λ_ADC (k_ADC=k_{Br 2}=k_{Br 1}) and then χ _th=0.

Figures 4(a) and (4b) show the dispersion diagrams for the particular case where the Bragg wavelength coincides with the initial phase matching wavelength. Figure 4(a) corresponds to the sub-case where the Bragg grating is located on the waveguide with higher index n_{g 1} >n_{g 2}. As is seen in this first sub-case, co-directional phase matching is systematically achieved for two frequencies whatever the Bragg coupling coefficient is. One frequency lies below the BG band gap while the other lies above this gap. Figure 4b corresponds to the second sub-case where the Bragg grating is located on the waveguide with lower index n_{g 1} <n_{g 2}. Co-directional coupling transmission is prohibited in this sub-case whatever the Bragg coupling coefficient is. As is seen in Fig. 4b, the original intersection between the dispersion curves of the two waveguides at ω_{Br 1}=ω_ADC disappears as soon as a gap is created around ω_{Br 1}, i.e. as soon as the value of the BG coupling coefficient χ ₁ is non zero. Actually, this behavior can be predicted from the analytical expression of χ _d as shown in what follows.

 

Fig. 4 Dispersion curves of the uniform waveguide (black) and the BG assisted waveguide for increasing values of the coupling coefficient χ₁ (from blue to light green) when ω_ADC=ω_Br1 (a) n_{g 1}>n_{g 2} (b) n_{g 1}<n_{g 2}. Insets show enlarged views in the vicinity of the phase matching wavelengths.

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The upper limit χ _d for co-directional coupling in the sub-case n_{g 1} <n_{g 2} can be derived from Eq. (2) with the same assumptions as those used to derive Eq. (4) (see Appendix B):

As for χ _th, the presence of two BGs (χ ₂≠0) tends to increase χ _d. Besides, χ_d depends on the spectral separation (λ_{Br 1}-λ_ADC) in a similar way as χ _th. The value of χ _d increases with the spectral separation. It cancels for λ_{Br 1}=λ_ADC since λ_{Br 1}=λ_{Br 2} and then k_{Br 1}=k_{Br 2} at the initial phase matching wavelength. It is worthwhile noticing that for χ ₂=0, the ratio χ _d/χ _th is independent of the spectral separation (λ_{Br 1}-λ_ADC). It is uniquely determined by the group indexes of the waveguides that form the ADC:

3. Coupled mode approach modeling

A coupled mode theory (CMT) approach has been used to verify the assertions inferred from the phase matching analysis of Section 2 and to explore the BGADC behavior in detail. For this purpose, the investigated device was schematized as shown in Fig. 5 . The schematized device consists of a five-layer structure (from N₁ to N₅) with two parallel slab waveguides surrounded by claddings. The waveguide in the upper part bears a double side BG of p periods with a total length L=pΛ. Both waveguides are assumed to be single mode. The dashed lines traced for the BG assisted waveguide indicate the original waveguide width without grating modulation. This width is in turn used to determine the effective index and the propagation constant β ₁=2πn ₁/λ, which is introduced in the system of coupled mode equations.

 

Fig. 5 Schematic representation of the Bragg grating assisted coupler with rectangular grating profile.

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A rectangular grating profile is assumed in such a way that the coupled-mode equations lead to analytical solutions. Following the approach developed in [23–27], the device is decomposed along the propagation axis z in a series of parallel waveguide segments of lengths Λ₊ and Λ_- (Λ = Λ₊ + Λ_-), delimited by the grating corrugations. Since all the coupling matrices are independent of z within each section, the coupled-mode equations can be solved exactly. To define the elements of the coupling matrices, we use an approach based on individual waveguide modes. The detailed description of the method can be found in [1].

In what follows, the developed CMT approach is applied to analyze the BGADC behavior depending on whether co-directional phase matching requires a threshold condition on the BG coupling coefficient or not. Let us notice, however, that the threshold values given in Eqs. (4-6) must be now multiplied by a factor of π/2 to account for the rectangular shape of the grating (Fig. 5).

4. Dual band operation with threshold condition for co-directional phase matching

When a threshold condition exists on χ ₁ for a dual band operation (i.e. for a co-directional coupling at two frequencies), it is of particular interest to compare the spectral responses of the BGADC below and above this threshold. Figures 6 and 7 show the spectral dependence of the co-directional coupling drop-port exchange transmission T ^× for several values of χ ₁ (with χ ₂=0) and for the two cases n_{g 1}>n_{g 2} and n_{g 1}<n_{g 2}, respectively. In both cases, the initial phase matching wavelength of the ADC is set to 1.55µm. To avoid undesirable reflection for optimal device operation, the incident light is injected in the uniform waveguide 2 (Fig. 5).

 

Fig. 6 Drop-port exchange transmission spectra calculated from the CMT model for several values of the coupling coefficient χ₁ when n_{g 1}>n_{g 2}. (a) ω_ADC<ω_Br1 (λ_ADC>λ_Br1) (b) ω_ADC>ω_Br1 (λ_ADC<λ_Br1).

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Fig. 7 Drop-port exchange transmission spectra calculated from the CMT model for several values of the coupling coefficient χ₁ when n _g1<n _g2 (a) ω_ADC<ω_Br1 (λ_ADC>λ_Br1) (b) ω_ADC>ω_Br1 (λ_ADC<λ_Br1).

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4.1 Bragg grating on the waveguide with higher group index (ng1>ng2)

For illustrating the case where ω_ADC<ω_Br1 (λ_ADC>λ_Br1) (Fig. 6(a)), the Bragg wavelength λ_Br1 is set to 1.535µm while n_{eff 1}=n_{eff 2}=3.26 and ∂n _{eff 1}/∂λ=–0.5µm ^–1; ∂n _{eff 2}/∂λ=–0.25µm ^–1. The latter values approximately correspond to the maximal asymmetry that can be achieved for a BGADC made of two vertically coupled InGaAsP/InP waveguides with different alloy compositions [22–24]. The values of χ ₁ chosen in the subsequent examples are: 0.5, 3, 5, and 10 χ _th. According to Eq. (5) with the correction for rectangular grating modulation, χ _th=0.027µm^–1. The grating length is L=1mm.

The transmission spectra of Fig. 6(a) confirm that below threshold (χ ₁=0.5χ _th), there is only one transmission band with Δλ _3dB=5nm bandwidth corresponding to the initial phase matching condition (λ_ADC=1.55 µm). The drop-port exchange transmission T^× near λ_Br1 is quite low. Its amplitude is of the same order as those of secondary transmission peaks. Above threshold (χ ₁=3χ _th), a narrow peak of Δλ _3dB=0.9nm bandwidth appears at a frequency (wavelength) slightly higher (smaller) than ω_Br1 (λ_Br1). The wavelength selectivity is thus enhanced by a factor of 5.5 as compared to the initial transmission band. However the amount of power transferred to the drop-port is not total in this case. The transmission level at maximum is around 79%. Despite the fact that the incident light is injected into the uniform waveguide, a fraction of it is reflected into this guide via an indirect mechanism involving Bragg reflection combined with co-directional coupling [1]. The transmission band associated to the initial phase matching condition (λ~λ _ADC) still exists, but its position is shifted toward longer wavelengths compared to the situation below threshold. Its width slightly narrows (Δλ_3dB=3.8nm), but remains several times larger then that of the transmission band in the vicinity of λ_Br1.

As χ ₁ is increased far above threshold (χ ₁=5χ _th, χ ₁=10χ _th), the BG induced transmission band progressively broadens and shifts to smaller wavelengths. Its amplitude grows with a tendency to approach some asymptotic limit. The transmission band associated to initial phase matching has a completely opposite behavior. Its bandwidth slightly narrows while its position shifts to longer wavelengths with the increase of χ ₁. At the same time, its amplitude is slightly decreasing. The asymptotic limits of the transmission amplitude and bandwidth are actually the same for the two bands when χ ₁ becomes infinitely large. This will be illustrated in Section 5 (Fig. 8(a) ).

 

Fig. 8 Drop-port exchange transmission spectra calculated from the CMT model for several values of the coupling coefficient χ₁ when ω_ADC=ω_Br1. (a) n _g1>n _g2, (b) n _g1<n _g2.

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For the case where ω_ADC>ω_Br1 (λ_ADC<λ_Br1) (Fig. 6(b)), the Bragg wavelength is set to 1.565µm in order to keep the same value of χ _th (0.027µm^–1). The other BGADC parameters are left unchanged. In agreement with the phase matching analysis of Section 2 (Fig. 2), the transmission band associated to the initial phase condition is now located at a frequency (wavelength) above ω_ADC (below λ_ADC) while the BG induced transmission band is located at a frequency (wavelength) below ω_ADC (above λ_ADC). Globally, the transmission spectra of Fig. 6(b) and those of Fig. 6(a) are symmetric with respect to λ_ADC. The evolutions of the two bands with χ ₁ in Fig. 6(b) can then be easily understood. At this stage, it should be mentioned that the dual band operation reported in Figs. (6a) and (6b) is not incompatible with demultiplexing applications, where single wavelength channel selection is required. A proper adjustment of the frequency difference |λ_ADC -λ_Br1| has to be made in such a way that the resulting spacing between the two bands be greater than the free spectral range (FSR) required for the chosen application.

4.2 Bragg grating on the waveguide with lower group index (n_g1<n_g2)

For the case where ω_ADC<ω_Br1 (λ_ADC>λ_Br1) (Fig. 7(a)), the Bragg wavelength is set to 1.535µm while n_{eff 1}=n_{eff 2}=3.26 and ∂n _{eff 1}/∂λ=–0.25µm ^–1; ∂n _{eff 2}/∂λ=–0.5µm ^–1. The BGADC has actually the same parameters as previously (n _g1>n _g2; ω_ADC<ω_Br1) except for the fact that the BG is now located on the waveguide with lower group index. The values of χ ₁ considered in the following examples are: 0.5, 1.7, 2.3, and 3χ _th where χ _th=0.022µm⁻¹. The example χ₁=2.3χ _th corresponds to the situation of tangential contact between dispersion curves. The grating length is L=1mm.

The transmission spectra of Fig. 7(a) show that below threshold (χ ₁=0.5χ _th), there is only one transmission band (Δλ _3dB=5nm) corresponding to the initial phase matching condition. The drop-port exchange transmission T^× in the vicinity of λ_Br is of the same order as those of the secondary peaks. Above threshold (χ ₁=1.5χ _th), in agreement with the phase matching analysis of Section 2, a narrow (Δλ _3dB=0.6nm) co-directional transmission band induced by the BG appears below the band gap (at wavelengths longer than the Bragg wavelength). The transmission level is 59%, which is much lower than that of the initial transmission band. Part of the light is back reflected into the input waveguide via Bragg reflection combined with co-directional coupling. The transmission band associated to the initial phase matching condition is shifted to higher frequencies, then approaching the Bragg frequency.

When χ ₁ =χ _d, the two bands merge into a single transmission band whose width (Δλ _3dB=9nm) is significantly larger than that of the initial band. Above the limit χ ₁=χ _d, the phase matching condition is no longer satisfied, and the drop-port exchange transmission T^× almost collapses. It is worthwhile noticing that the upper limit for co-directional coupling χ _d is presently much lower than that corresponding to a system of two BG coupled waveguides without initial matching condition [1]. It can be easily attained in practical situations. Indeed, for a rectangular grating profile, the relation between the BG coupling coefficient χ ₁ and the waveguide effective index modulation Δn_{eff 1} is:

The waveguide index modulation corresponding to χ _d is in our example 2.5⋅10⁻². Such a value is quite achievable in the InGaAsP/InP and Silicon on Insulator (SOI) waveguide systems. This shows the strong influence of the grating location. For the same BG coupling strength, the device behavior is drastically different according as the grating is placed on the low or high group index waveguide.

Figure 7(b) shows the transmission spectra for ω_ADC>ω_Br1 (λ_ADC<λ_Br1). The Bragg wavelength is set to 1.565µm in such a way that the threshold coupling strength is the same as in the previous case ω_ADC<ω_Br1 (λ_ADC>λ_Br1). Other BGADC parameters remain unchanged. In agreement with the phase matching analysis of section 2 (Fig. 3), both transmission bands are now located at frequencies (wavelengths) higher than ω_Br1 (smaller than λ_Br1). Globally, the transmission spectra of Fig. 7(b) and those of Fig. 7(a) are symmetric with respect to λ_ADC=1.55µm. For the rest, the behavior is similar to that described above for λ_ADC>λ_Br1. The evolutions of the two bands with χ ₁ in Fig. 7(b) can then be easily understood.

Clearly, results of Fig. 7 show that the configuration with the Bragg grating located on the waveguide with low group index is less suitable for practical applications than that with the Bragg grating on the high group index waveguide.

5. Dual band operation without threshold condition for co-directional phase matching

As previously shown in section 2 (Fig. (4)), particular behaviors of the BGADC are predicted when ω_ADC=ω_Br1 and more generally, when ω_ADC≈ω_Br1. Indeed, as explained in section 2, the threshold for co-directional phase matching tends to vanish in such a case if n _g1>n _g2. In contrast, co-directional phase-matching is forbidden if n _g1<n _g2.

5.1 Bragg grating on the waveguide with higher group index (n_g1>n_g2)

Here we adopt a reference coupling coefficient χ _ref, whose value (0.027µm⁻¹) is identical to that previously used for χ _th. The phase matching wavelength between the two guides in the absence of the BG is set to 1.55µm, which is also the value of the Bragg wavelength. The refractive index parameters of the BGADC are the same as those previously used in the n _g1>n _g2 case in Section 4: n_{eff 1}=n_{eff 2}=3.26; ∂n _{eff 1}/∂λ=^–0.5µm ^–1; ∂n _{eff 2}/∂λ=^–0.25µm ^–1. The values of χ ₁ chosen for the illustrations (Fig. 8(a)) are: 0, 0.5, 5, and 10 χ _ref. The grating length is set to be L=0.5mm for a better display of the spectral features.

For small values of the coupling coefficient (χ ₁=0.5χ _ref), the initial transmission band at ω_ADC=ω_Br1 starts splitting into two quasi-identical and symmetrical bands. The width of each of the two bands is approximately half that of the initial band. An increase of the coupling coefficient (χ ₁=5χ _ref, χ ₁=10χ _ref) leads to an increase of the spectral separation between these symmetrical bands. Regardless of the χ ₁ value, the level of the back-reflection losses is quite low (~5%), and the transmission to the drop-port approximately reaches 95%.

This thresholdless mode of operation can be viewed as a particular case where the separation between the initial phase matching frequency and the Bragg frequency is infinitely smaller than the spectral separation between the two transmission bands. In other words, this mode can be viewed as the asymptotic limit of standard modes of operation with ω_ADC≠ω_Br1 when the value of the coupling coefficient tends to be infinitely larger than the threshold value. This explains the evolutions of the two bands in Figs. 6(a) and 6(b) for high values of χ ₁: The separation between transmission bands tends to be very large, the widths of the two bands tend to be equal, and the two transmission levels at maximum tend to be close to unity.

5.2 Bragg grating on the waveguide with lower group index (n_g1<n_g2)

For the case where n _g1<n _g2, the reference coupling coefficient χ _ref is set equal to χ _th=0.022µm⁻¹ as in the preceding examples shown in Section 4 for n _g1<n _g2. The initial phase matching wavelength and the Bragg wavelength are set to be equal to 1.55µm. The refractive index parameters of the ADC are: n_{eff 1}=n_{eff 2}=3.26; ∂n _{eff 1}/∂λ=–0.25µm ^–1; ∂n _{eff 2}/∂=–0.5µm ^–1. In other words, the ADC waveguide structure is the same as in the previous case for n _g1>n _g2 except for the fact that the BG is now located on the waveguide with lower group index. The values of χ ₁ chosen for graphic illustrations (Fig. 8(b)) are: 0, 0.5, 2, and 4χ _ref. The grating length is set to be L=0.5mm.

For small values of the coupling coefficient (χ ₁=0.5χ _ref), the initial transmission band (without BG) splits into two symmetrical bands around the Bragg frequency (Fig. 8(b)). The amplitude of these two bands, though lower than the initial band, is still important. This seems to be in contradiction with the phase matching analysis of Section 2 (Fig. 4(b)), which predicts the impossibility of co-directional coupling. This contradiction stems from the fact that for a small coupling coefficient (χ ₁=0.5χ _ref) and a moderate grating length, the grating strength (i.e. the product (χ ₁L)) is not sufficient to benefit from the maximum dispersion variation available with the BG. Numerical examples (not shown here) confirm that for a longer grating and the same coupling coefficient, the transmission drops to the level of secondary peaks. Figure 8(b) shows that for larger coupling coefficients (χ ₁=2χ _ref, χ ₁=4χ _ref) and the moderate grating length L=0.5mm, the transmission level is very low, of the order of the secondary peaks.

6. Summary and conclusions

The aim of the present work was to investigate the possibilities of designing highly wavelength-selective directional couplers with the use of an artificial dispersion created by material structuring. Our approach has been based on the use of the dispersive properties of Bragg gratings (BG) operated near their own photonic band gaps. Unlike previous investigations in the case of asymmetric waveguide couplers with low differential dispersion between the two guides [1], this paper has been directed to asymmetric waveguide couplers with high differential dispersion between the guides.

It has been shown that in most cases, the improvement of the coupler selectivity requires the fulfillment of a threshold condition on the BG coupling coefficient. The presence of BG(s) leads in turn to the appearance of two transmission bands instead of one. The wavelength selectivity associated to the BG induced transmission band can be much higher than that obtained in the absence of BG(s). The use of a simplified model with the assumption of a linear wavelength dispersion for the waveguides effective indexes has allowed an analytical expression of the BG coupling threshold as well as a general description of the coupler behavior. It has been shown that the optimal BGADC operation was achieved with only one Bragg grating distributed along one of the two waveguides.

The performed analysis has revealed that the device behavior critically depends on whether the BG is located on the waveguide with lower or higher effective index. Operating the coupler with the BG on the low group index waveguide turns to be not suitable for most practical applications. Conversely, operating the coupler with the BG on the high group index waveguide leads to a significant improvement of the wavelength selectivity.

Theoretical predictions from the analytical model have been successfully verified using a couple mode theory (CMT) four-wave model. Results of CMT modeling have shown the possibility of obtaining a 0.9 nm bandwidth at telecommunication wavelengths with a 1mm long device and only 1dB of reflection loss penalty. This selectivity is a factor of 5.5 higher than that of the state-of-the-art approach of vertically coupled InGaAsP/InP waveguides with different alloy compositions. The selectivity could still be improved to a ~0.65nm bandwidth by using a larger spectral spacing between the Bragg wavelength and the initial phase matching wavelength. Further selectivity enhancement beyond the above value can also be achieved with an increase of the device length. There will not be any additional loss, provided that the dispersion variation induced by the BG has reached its limit. Besides, the existence of a second, nearby, coupler transmission band will not be a problem since the frequency separation between these two bands can be set large enough to prevent undesired overlap. Regarding polarization aspects, which are critical for any planar waveguide component, solutions for polarization insensitivity reported in [28–30] can be adapted to the present coupler configuration. This subject actually merits a specific treatment, which is out of the scope of this paper

Finally, it has been shown that under particular circumstances, dual band operation can be achieved without any threshold condition. Indeed, when the Bragg wavelength (nearly) coincides with the initial phase matching wavelength, the two transmission bands have almost identical widths and amplitudes. This situation is radically different from those previously analyzed in the case of initially mismatched waveguides [1], where only one transmission band could be obtained under some threshold condition. Here, the two band centers are symmetrical with respect to the Bragg wavelength, and the spectral separation between them is adjustable with the BG coupling strength. The width of each band is approximately two times narrower than that of the initial transmission band in the absence of the BG. The additional loss level is quite low in such a way that the transmission to the drop-port is approximately 95% regardless of the BG coupling coefficient. Such a dual band transmission coupler is expected to offer new functionalities for wavelength demultiplexing applications. A possible application to “Fiber To The Home” networks would be the extraction of two career wavelengths with a single coupler device. The achievement of a dual mode laser emission with tunability of the two emitted wavelengths could be another opportunity.