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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 2 — Jan. 17, 2011
  • pp: 1246–1259
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Dual transmission band Bragg grating assisted asymmetric directional couplers

Anatole Lupu, Kamal Muhieddine, Eric Cassan, and Jean-Michel Lourtioz  »View Author Affiliations


Optics Express, Vol. 19, Issue 2, pp. 1246-1259 (2011)
http://dx.doi.org/10.1364/OE.19.001246


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Abstract

The use of artificial dispersion by material structuring is investigated for the design of highly wavelength selective directional couplers. Systems of two highly asymmetric coupled waveguides are considered with the artificial dispersion created by distributed Bragg gratings (BGs) operated near photonic band gap. It is shown that even in the case of an asymmetrical directional coupler with initially phase matched waveguides, the achievement of high wavelength 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 one of these bands is much higher than that obtained in the absence of BG(s). It is also shown that under particular circumstances, dual band operation can be achieved without threshold condition. The directional coupler then exhibits two transmission bands with approximately the same width and a very low level of insertion losses. Such a dual band transmission coupler is expected to offer new functionalities for wavelength demultiplexing applications.

© 2011 OSA

1. Introduction

In a recent paper [1

1. K. Muhieddine, A. Lupu, E. Cassan, and J.-M. Lourtioz, “Proposal and analysis of narrow band transmission asymmetric directional couplers with Bragg grating induced phase matching,” Opt. Express 18(22), 23183–23195 (2010). [CrossRef] [PubMed]

], 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

1. K. Muhieddine, A. Lupu, E. Cassan, and J.-M. Lourtioz, “Proposal and analysis of narrow band transmission asymmetric directional couplers with Bragg grating induced phase matching,” Opt. Express 18(22), 23183–23195 (2010). [CrossRef] [PubMed]

] 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

2. C. Wu, C. Rolland, N. Puetz, R. Bruce, K. D. Chik, and J. M. Xu, “A vertically coupled InGaAsP/InP directional coupler filter of ultranarrow bandwidth,” IEEE Photon. Technol. Lett. 3(6), 519–521 (1991). [CrossRef]

4

4. S.-K. Han, R. V. Ramaswamy, and R. F. Tavlykaev, “Highly asymmetrical vertical coupler wavelength filter in InGaAlAs/InP,” Electron. Lett. 33, 30–31 (1999).

].

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

1. K. Muhieddine, A. Lupu, E. Cassan, and J.-M. Lourtioz, “Proposal and analysis of narrow band transmission asymmetric directional couplers with Bragg grating induced phase matching,” Opt. Express 18(22), 23183–23195 (2010). [CrossRef] [PubMed]

]. 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

8. P. Yeh and H. F. Taylor, “Contradirectional frequency-selective couplers for guided-wave optics,” Appl. Opt. 19(16), 2848–2855 (1980). [CrossRef] [PubMed]

19

19. A. Yesayan and R. Vallée, “Zero backreflection condition for a grating-frustrated coupler,” J. Opt. Soc. Am. B 20(7), 1418–1426 (2003). [CrossRef]

], 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

20. N. Imoto, “An analysis for contradirectional-coupler-type optical grating filters,” J. Lightwave Technol. 3(4), 895–900 (1985). [CrossRef]

, 21

21. J. E. Sipe, L. Poladian, and C. M. de Sterke, “Propagation through nonuniform grating structures,” J. Opt. Soc. Am. A 11(4), 1307–1320 (1994). [CrossRef]

] 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

22. A. Yariv and M. Nakamura, “Periodic structures for integrated optics,” IEEE J. Quantum Electron. 13(4), 233–253 (1977). [CrossRef]

] 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 neff:

βeq=(πΛ±(kneff(k)πΛ)χ2)
(1)

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:

πΛ1±(kneff1(k)πΛ1)2χ12=πΛ2±(kneff2(k)πΛ2)2χ22
(2)

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 Λ12=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:
k=ng12kBr12ng22kBr22(χ12χ22)(ng12kBr1ng22kBr2)±ng12ng22(kBr1kBr2)2+(χ12χ22)(ng12ng22)
(3)
where ng1, ng2 are the waveguides group indexes and kBr1=2π/λBr1, kBr2=2π/λBr2 are the Bragg wave-vectors for the two guides, respectively. As seen, expression (3) depends on the difference of the squares of the coupling coefficients (χ12χ22), but not on χ 1 and χ 2 separately. Therefore, the analysis of phase-matching conditions can be reduced to the case where there is only one BG. A further inspection of Eq. (3) shows that the wave-vector solutions differ according as ng1 is smaller or larger than ng2. Each case has then to be separately treated. In what follows, we can arbitrarily assume that χ 1≠0 and χ 2 = 0.

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

Figures 2(a)
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 ng1>ng2. (a) ωADCBr1 (b) ωADCBr1. Insets show enlarged views in the vicinity of the phase matching wavelengths.
and 2(b) show the waveguide dispersion characteristics, ω = ω(βeq), calculated for ng1>n g2. In Fig. 2(a), the grating period Λ is chosen in such a way that the Bragg frequency ωBr1 is larger than ωADC, which is the phase-matching frequency in the absence of the BG. The reverse situation (ωADCBr1) 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.

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 ωBr1, 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 ωBr1, 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 χ1=χth 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 χth.

For the situation depicted in Fig. 2(b)ADCBr1), phase matching due to the BG occurs at a frequency below ωBr1. Except for that, the behavior is similar to that previously encountered for ωADCBr1. A minimum coupling coefficient χth 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)
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 ng1<ng2. (a) ωADCBr1 (b) ωADCBr1. Insets show enlarged views in the vicinity of the phase matching wavelengths.
and 3(b) show the waveguide dispersion curves when the BG is placed on the waveguide with lower group index (ng1<ng2). Figure 3(a) corresponds to the situation where ωADCBr1. In this situation, dispersion curves of the two guides only intersect at frequencies below ωBr1. 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).

At a certain value of the coupling coefficient χ1=χth, 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: χth<χ1<χd. 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 ωADCBr1 (Fig. 3(b)), the behavior is similar to that previously described for ωADCBr1 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 (χthandχth) can be derived from Eq. (2) by locally approximating the dispersion relations of the guides with linear functions and assuming that: χ 1χ 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 χthandχth, i.e. whether n g1>n g2 or n g1<n g2:

χth=ng1|kBr1kBr2|+ng1ng2χ2
(4)

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 λADC is explicitly included:

χth=ng1πΛ((λADCλBr1)(dneff1dλdneff2dλ)neff1(λBr1)(neff1(λBr1)+(λADCλBr1)(dneff1dλdneff2dλ)))+ng1ng2χ2
(5)

It then appears that the threshold value χ th for co-directional phase matching increases with the separation between λADC and the BG wavelength λBr 1. Conversely, χ th decreases when λBr 1 gets closer to λADC, and even cancels when λBr 1=λADC. 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 λADC. The equality λBr 1=λADC directly leads to λBr 2=λBr 1=λADC (kADC=kBr 2=kBr 1) and then χ th=0.

Figures 4(a)
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 ωADCBr1 (a) ng 1>ng 2 (b) ng 1<ng 2. Insets show enlarged views in the vicinity of the phase matching wavelengths.
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 ng 1 >ng 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 ng 1 <ng 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 ωBr 1ADC disappears as soon as a gap is created around ωBr 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.

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

χd=ng12ng22ng12ng22(kBr1kBr2)2+χ22
(6)

As for χ th, the presence of two BGs (χ 2≠0) tends to increase χ d. Besides, χd depends on the spectral separation (λBr 1-λADC) in a similar way as χ th. The value of χ d increases with the spectral separation. It cancels for λBr 1=λADC since λBr 1=λBr 2 and then kBr 1=kBr 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 (λBr 1-λADC). It is uniquely determined by the group indexes of the waveguides that form the ADC:

χdχth=ng22ng12ng22
(7)

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

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

23. R. März and H. P. Nolting, “Spectral properties of asymmetrical optical directional couplers with periodic structures,” Opt. Quantum Electron. 19(5), 273–287 (1987). [CrossRef]

27

27. N. Izhaky and A. Hardy, “Analysis of grating-assisted backward coupling employing the unified coupled-mode formalism,” J. Opt. Soc. Am. A 16(6), 1303–1311 (1999). [CrossRef]

], 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

1. K. Muhieddine, A. Lupu, E. Cassan, and J.-M. Lourtioz, “Proposal and analysis of narrow band transmission asymmetric directional couplers with Bragg grating induced phase matching,” Opt. Express 18(22), 23183–23195 (2010). [CrossRef] [PubMed]

].

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
Fig. 6 Drop-port exchange transmission spectra calculated from the CMT model for several values of the coupling coefficient χ1 when ng 1>ng 2. (a) ωADCBr1ADCBr1) (b) ωADCBr1ADCBr1).
and 7
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) ωADCBr1ADCBr1) (b) ωADCBr1ADCBr1).
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 ng 1>ng 2 and ng 1<ng 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).

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

For illustrating the case where ωADCBr1ADCBr1) (Fig. 6(a)), the Bragg wavelength λBr1 is set to 1.535µm while neff 1=neff 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

22. A. Yariv and M. Nakamura, “Periodic structures for integrated optics,” IEEE J. Quantum Electron. 13(4), 233–253 (1977). [CrossRef]

24

24. R. R. A. Syms, “Improved coupled mode theory for codirectionally and contradirectionally coupled waveguide arrays,” J. Opt. Soc. Am. A 8(7), 1062–1069 (1991). [CrossRef]

]. 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 (λADC=1.55 µm). The drop-port exchange transmission T× near λBr1 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 ωBr1Br1). 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

1. K. Muhieddine, A. Lupu, E. Cassan, and J.-M. Lourtioz, “Proposal and analysis of narrow band transmission asymmetric directional couplers with Bragg grating induced phase matching,” Opt. Express 18(22), 23183–23195 (2010). [CrossRef] [PubMed]

]. 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 (Δλ3dB=3.8nm), but remains several times larger then that of the transmission band in the vicinity of λBr1.

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 ωADCBr1. (a) n g1>n g2, (b) n g1<n g2.
).

For the case where ωADCBr1ADCBr1) (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 ωADC (below λADC) while the BG induced transmission band is located at a frequency (wavelength) below ωADC (above λADC). Globally, the transmission spectra of Fig. 6(b) and those of Fig. 6(a) are symmetric with respect to λADC. 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 |λADCBr1| 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 (ng1<ng2)

For the case where ωADCBr1ADCBr1) (Fig. 7(a)), the Bragg wavelength is set to 1.535µm while neff 1=neff 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; ωADCBr1) 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 λBr 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 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

1. K. Muhieddine, A. Lupu, E. Cassan, and J.-M. Lourtioz, “Proposal and analysis of narrow band transmission asymmetric directional couplers with Bragg grating induced phase matching,” Opt. Express 18(22), 23183–23195 (2010). [CrossRef] [PubMed]

]. 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 Δneff 1 is:

Δneff1=λBr1χ1π
(8)

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.

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 ωADCBr1 and more generally, when ωADC≈ωBr1. 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 (ng1>ng2)

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: neff 1=neff 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 ωADCBr1 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 ωADC≠ωBr1 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 (ng1<ng2)

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

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

28. S. François, S. Fouchet, N. Bouadma, A. Ougazzaden, M. Carré, G. Hervé-Gruyer, M. Filoche, and A. Carenco, “Polarization independent filtering in a grating assisted horizontal directional coupler,” IEEE Photon. Technol. Lett. 7(7), 780–782 (1995). [CrossRef]

30

30. Q. Guo and W.-P. Huang, “Polarisation-independent optical filters based on co-directional phase-shifted grating-assisted couplers: theory and design,” IEEE Proc. Optoelectron. 143(3), 173–177 (1996). [CrossRef]

] 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

1. K. Muhieddine, A. Lupu, E. Cassan, and J.-M. Lourtioz, “Proposal and analysis of narrow band transmission asymmetric directional couplers with Bragg grating induced phase matching,” Opt. Express 18(22), 23183–23195 (2010). [CrossRef] [PubMed]

], 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.

Appendix A: Local linear approximation solutions Eq. (3)

To solve Eq. (2), we locally approximate the dispersion of each waveguide by a linear function. The waveguide effective index expresses as:

neff(λ)=neff(λ0)+(λλ0)neffλ=ng+λneffλ
(A1)

neffλ12Λ=neffλnλBrλBr=ngλBr
(A2)

Eq. (3) is then derived from Eqs. (2), (A1) and (A2) using straightforward but somewhat lengthy calculations.

Appendix B: Threshold condition in Eq. (4) and upper limit of χ1 in Eq(6)

As can be seen in Figs. 2 and 3, the threshold for co-directional coupling induced by the BG is reached when the real parts of the propagation constants are both equal to π/Λ. The wave-vector satisfying the phase matching condition can then be directly determined from Eq. (2):

|2πλneff1(λ)πΛ||χ1|=0
(B1)
|2πλneff2(λ)πΛ||χ2|=0
(B2)

Eqs. (B1) and (B2) represent a system of equations where the unknown quantities are the phase matching wavelength and the threshold value for the coupling coefficient χ 1=χ th. The phase matching wave-vector and then the phase matching wavelength can actually be determined from Eq. (3) when χ 1 and χ 2 are known.

Assuming a linear approximation for the dispersion characteristics (Eq. (A1)), the system of Eqs. (B1)-(B2) also becomes linear. Then, we obtain:

χ1=|πΛ(1ng1ng2)2π(neff1λng1ng2neff2λ)|+χ2ng1ng2
(B3)

The expression for the threshold χ 1 coefficient (Eq. (4)) is derived in a straightforward manner from Eq. (B3) with the help of Eq. (A2).

In order to explicitly introduce the spectral separation between the wavelength and the initial phase matching wavelength, it is convenient to express Eq. (4) in a slightly different form:

χ1=|ng1πΛ(1neff1(λBr1)1neff2(λBr2))|+χ2ng1ng2
(B4)

Since for λ=λ ADC the effective indexes of both waveguides are equal, the following relation is easily established:

neff1(λADC)=neff1(λBr1)(λBr1λADC)neff1λ=neff2(λBr1)(λBr1λADC)neff2λ=neff2(λADC)
(B5)

Eq. (5) is then obtained by substituting neff 1Br1) and neff 12Br1) from Eq. (B5) into Eq. (B4). Besides, the upper limit for co-directional coupling χ 1=χ d is determined by assuming a tangential contact between the dispersion curves of the two guides. In that case, there is only one solution for Eq. (3). Eq. (6) in Section 2 is then readily obtained from elementary algebra.

References and links

1.

K. Muhieddine, A. Lupu, E. Cassan, and J.-M. Lourtioz, “Proposal and analysis of narrow band transmission asymmetric directional couplers with Bragg grating induced phase matching,” Opt. Express 18(22), 23183–23195 (2010). [CrossRef] [PubMed]

2.

C. Wu, C. Rolland, N. Puetz, R. Bruce, K. D. Chik, and J. M. Xu, “A vertically coupled InGaAsP/InP directional coupler filter of ultranarrow bandwidth,” IEEE Photon. Technol. Lett. 3(6), 519–521 (1991). [CrossRef]

3.

B. Liu, A. Shakouri, P. Abraham, Y. J. Chiu, S. Zhang, and J. E. Bowers, “Fused InP–GaAs Vertical Coupler Filters,” IEEE Photon. Technol. Lett. 11, 93–95 (1995).

4.

S.-K. Han, R. V. Ramaswamy, and R. F. Tavlykaev, “Highly asymmetrical vertical coupler wavelength filter in InGaAlAs/InP,” Electron. Lett. 33, 30–31 (1999).

5.

C. Bornhold, F. Kappe, R. Müller, H.-P. Nolting, F. Reier, R. Stenzel, H. Venghaus, and C. M. Weinert, “Meander coupler, a novel wavelength division multiplexer/demultiplexer,” Appl. Phys. Lett. 57(24), 2517–2519 (1990). [CrossRef]

6.

A. Lupu, P. Win, H. Sik, P. Boulet, M. Carre, J. Landreau, S. Slempkes, and A. Carenco, “Tunable filter with box like spectral response for 1.28/1.32 µm duplexer application,” Electron. Lett. 35(2), 174–175 (1999). [CrossRef]

7.

A. Lupu, H. Sik, A. Mereuta, P. Boulet, M. Carré, S. Slempkes, A. Ougazzaden, and A. Carenco, “Three-waveguides two-grating codirectional coupler for 1.3-/1.3+/1.5µm demultiplexing in transceiver,” Electron. Lett. 36(24), 2030–2031 (2000). [CrossRef]

8.

P. Yeh and H. F. Taylor, “Contradirectional frequency-selective couplers for guided-wave optics,” Appl. Opt. 19(16), 2848–2855 (1980). [CrossRef] [PubMed]

9.

M. S. Whalen, M. D. Divino, and R. C. Alferness, “Demonstration of a narrowband Bragg-reflection filter in a single-mode fibre directional coupler,” Electron. Lett. 22(12), 681–682 (1986). [CrossRef]

10.

R. R. A. Syms, “Optical directional coupler with a grating overlay,” Appl. Opt. 24(5), 717–726 (1985). [CrossRef] [PubMed]

11.

R. März and H. P. Nolting, “Spectral properties of asymmetrical optical directional couplers with periodic structures,” Opt. Quantum Electron. 19(5), 273–287 (1987). [CrossRef]

12.

L. Dong, P. Hua, T. A. Birks, L. Reekie, and P. S. J. Russell, “Novel add–drop filters for wavelength division multiplexing optical fiber systems using a Bragg grating assisted mismatched coupler,” IEEE Photon. Technol. Lett. 8(12), 1656–1658 (1996). [CrossRef]

13.

S. S. Orlov, A. Yariv, and S. Van Essen, “Coupled-mode analysis of fiber-optic add drop filters for dense wavelength-division multiplexing,” Opt. Lett. 22(10), 688–690 (1997). [CrossRef] [PubMed]

14.

T. Erdogan, “Optical add–drop multiplexer based on an asymmetric Bragg coupler,” Opt. Commun. 157(1-6), 249–264 (1998). [CrossRef]

15.

I. Baumann, J. Seifert, W. Novak, and M. Sauer, “Compact all-fiber add-drop-multiplexer using fiber Bragg gratings,” IEEE Photon. Technol. Lett. 8(10), 1331–1333 (1996). [CrossRef]

16.

J.-L. Archambault, P. St. J. Russell, S. Barcelos, P. Hua, and L. Reekie, “Grating-frustrated coupler: a novel channel-dropping filter in single-mode optical fiber,” Opt. Lett. 19(3), 180–182 (1994). [CrossRef] [PubMed]

17.

A.-C. Jacob-Poulin, R. Valle’e, S. LaRochelle, D. Faucher, and G. R. Atkins, “Channel-dropping filter based on a grating-frustrated two-core fiber,” J. Lightwave Technol. 18(5), 715–720 (2000). [CrossRef]

18.

A. Yesayan and R. Vallée, “Optimized grating-frustrated coupler,” Opt. Lett. 26(17), 1329–1331 (2001). [CrossRef]

19.

A. Yesayan and R. Vallée, “Zero backreflection condition for a grating-frustrated coupler,” J. Opt. Soc. Am. B 20(7), 1418–1426 (2003). [CrossRef]

20.

N. Imoto, “An analysis for contradirectional-coupler-type optical grating filters,” J. Lightwave Technol. 3(4), 895–900 (1985). [CrossRef]

21.

J. E. Sipe, L. Poladian, and C. M. de Sterke, “Propagation through nonuniform grating structures,” J. Opt. Soc. Am. A 11(4), 1307–1320 (1994). [CrossRef]

22.

A. Yariv and M. Nakamura, “Periodic structures for integrated optics,” IEEE J. Quantum Electron. 13(4), 233–253 (1977). [CrossRef]

23.

R. März and H. P. Nolting, “Spectral properties of asymmetrical optical directional couplers with periodic structures,” Opt. Quantum Electron. 19(5), 273–287 (1987). [CrossRef]

24.

R. R. A. Syms, “Improved coupled mode theory for codirectionally and contradirectionally coupled waveguide arrays,” J. Opt. Soc. Am. A 8(7), 1062–1069 (1991). [CrossRef]

25.

J. Hong and W. P. Huang, “Contra-directional coupling in grating-assisted devices,” J. Lightwave Technol. 10(7), 873–881 (1992). [CrossRef]

26.

A. Hardy, “A unified approach to coupled-mode phenomena,” IEEE J. Quantum Electron. 34(7), 1109–1116 (1998). [CrossRef]

27.

N. Izhaky and A. Hardy, “Analysis of grating-assisted backward coupling employing the unified coupled-mode formalism,” J. Opt. Soc. Am. A 16(6), 1303–1311 (1999). [CrossRef]

28.

S. François, S. Fouchet, N. Bouadma, A. Ougazzaden, M. Carré, G. Hervé-Gruyer, M. Filoche, and A. Carenco, “Polarization independent filtering in a grating assisted horizontal directional coupler,” IEEE Photon. Technol. Lett. 7(7), 780–782 (1995). [CrossRef]

29.

Y. Shibata, T. Tamamura, S. Oku, and Y. Kondo, “Coupling coefficient modulation of waveguide grating using sampled grating,” IEEE Photon. Technol. Lett. 6(10), 1222–1224 (1994). [CrossRef]

30.

Q. Guo and W.-P. Huang, “Polarisation-independent optical filters based on co-directional phase-shifted grating-assisted couplers: theory and design,” IEEE Proc. Optoelectron. 143(3), 173–177 (1996). [CrossRef]

OCIS Codes
(130.3120) Integrated optics : Integrated optics devices

ToC Category:
Integrated Optics

History
Original Manuscript: November 15, 2010
Revised Manuscript: December 29, 2010
Manuscript Accepted: January 2, 2011
Published: January 11, 2011

Citation
Anatole Lupu, Kamal Muhieddine, Eric Cassan, and Jean-Michel Lourtioz, "Dual transmission band Bragg grating assisted asymmetric directional couplers," Opt. Express 19, 1246-1259 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-2-1246


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References

  1. K. Muhieddine, A. Lupu, E. Cassan, and J.-M. Lourtioz, “Proposal and analysis of narrow band transmission asymmetric directional couplers with Bragg grating induced phase matching,” Opt. Express 18(22), 23183–23195 (2010). [CrossRef] [PubMed]
  2. C. Wu, C. Rolland, N. Puetz, R. Bruce, K. D. Chik, and J. M. Xu, “A vertically coupled InGaAsP/InP directional coupler filter of ultranarrow bandwidth,” IEEE Photon. Technol. Lett. 3(6), 519–521 (1991). [CrossRef]
  3. B. Liu, A. Shakouri, P. Abraham, Y. J. Chiu, S. Zhang, and J. E. Bowers, “Fused InP–GaAs Vertical Coupler Filters,” IEEE Photon. Technol. Lett. 11, 93–95 (1995).
  4. S.-K. Han, R. V. Ramaswamy, and R. F. Tavlykaev, “Highly asymmetrical vertical coupler wavelength filter in InGaAlAs/InP,” Electron. Lett. 33, 30–31 (1999).
  5. C. Bornhold, F. Kappe, R. Müller, H.-P. Nolting, F. Reier, R. Stenzel, H. Venghaus, and C. M. Weinert, “Meander coupler, a novel wavelength division multiplexer/demultiplexer,” Appl. Phys. Lett. 57(24), 2517–2519 (1990). [CrossRef]
  6. A. Lupu, P. Win, H. Sik, P. Boulet, M. Carre, J. Landreau, S. Slempkes, and A. Carenco, “Tunable filter with box like spectral response for 1.28/1.32 µm duplexer application,” Electron. Lett. 35(2), 174–175 (1999). [CrossRef]
  7. A. Lupu, H. Sik, A. Mereuta, P. Boulet, M. Carré, S. Slempkes, A. Ougazzaden, and A. Carenco, “Three-waveguides two-grating codirectional coupler for 1.3-/1.3+/1.5µm demultiplexing in transceiver,” Electron. Lett. 36(24), 2030–2031 (2000). [CrossRef]
  8. P. Yeh and H. F. Taylor, “Contradirectional frequency-selective couplers for guided-wave optics,” Appl. Opt. 19(16), 2848–2855 (1980). [CrossRef] [PubMed]
  9. M. S. Whalen, M. D. Divino, and R. C. Alferness, “Demonstration of a narrowband Bragg-reflection filter in a single-mode fibre directional coupler,” Electron. Lett. 22(12), 681–682 (1986). [CrossRef]
  10. R. R. A. Syms, “Optical directional coupler with a grating overlay,” Appl. Opt. 24(5), 717–726 (1985). [CrossRef] [PubMed]
  11. R. März and H. P. Nolting, “Spectral properties of asymmetrical optical directional couplers with periodic structures,” Opt. Quantum Electron. 19(5), 273–287 (1987). [CrossRef]
  12. L. Dong, P. Hua, T. A. Birks, L. Reekie, and P. S. J. Russell, “Novel add–drop filters for wavelength division multiplexing optical fiber systems using a Bragg grating assisted mismatched coupler,” IEEE Photon. Technol. Lett. 8(12), 1656–1658 (1996). [CrossRef]
  13. S. S. Orlov, A. Yariv, and S. Van Essen, “Coupled-mode analysis of fiber-optic add drop filters for dense wavelength-division multiplexing,” Opt. Lett. 22(10), 688–690 (1997). [CrossRef] [PubMed]
  14. T. Erdogan, “Optical add–drop multiplexer based on an asymmetric Bragg coupler,” Opt. Commun. 157(1-6), 249–264 (1998). [CrossRef]
  15. I. Baumann, J. Seifert, W. Novak, and M. Sauer, “Compact all-fiber add-drop-multiplexer using fiber Bragg gratings,” IEEE Photon. Technol. Lett. 8(10), 1331–1333 (1996). [CrossRef]
  16. J.-L. Archambault, P. St. J. Russell, S. Barcelos, P. Hua, and L. Reekie, “Grating-frustrated coupler: a novel channel-dropping filter in single-mode optical fiber,” Opt. Lett. 19(3), 180–182 (1994). [CrossRef] [PubMed]
  17. A.-C. Jacob-Poulin, R. Valle’e, S. LaRochelle, D. Faucher, and G. R. Atkins, “Channel-dropping filter based on a grating-frustrated two-core fiber,” J. Lightwave Technol. 18(5), 715–720 (2000). [CrossRef]
  18. A. Yesayan and R. Vallée, “Optimized grating-frustrated coupler,” Opt. Lett. 26(17), 1329–1331 (2001). [CrossRef]
  19. A. Yesayan and R. Vallée, “Zero backreflection condition for a grating-frustrated coupler,” J. Opt. Soc. Am. B 20(7), 1418–1426 (2003). [CrossRef]
  20. N. Imoto, “An analysis for contradirectional-coupler-type optical grating filters,” J. Lightwave Technol. 3(4), 895–900 (1985). [CrossRef]
  21. J. E. Sipe, L. Poladian, and C. M. de Sterke, “Propagation through nonuniform grating structures,” J. Opt. Soc. Am. A 11(4), 1307–1320 (1994). [CrossRef]
  22. A. Yariv and M. Nakamura, “Periodic structures for integrated optics,” IEEE J. Quantum Electron. 13(4), 233–253 (1977). [CrossRef]
  23. R. März and H. P. Nolting, “Spectral properties of asymmetrical optical directional couplers with periodic structures,” Opt. Quantum Electron. 19(5), 273–287 (1987). [CrossRef]
  24. R. R. A. Syms, “Improved coupled mode theory for codirectionally and contradirectionally coupled waveguide arrays,” J. Opt. Soc. Am. A 8(7), 1062–1069 (1991). [CrossRef]
  25. J. Hong and W. P. Huang, “Contra-directional coupling in grating-assisted devices,” J. Lightwave Technol. 10(7), 873–881 (1992). [CrossRef]
  26. A. Hardy, “A unified approach to coupled-mode phenomena,” IEEE J. Quantum Electron. 34(7), 1109–1116 (1998). [CrossRef]
  27. N. Izhaky and A. Hardy, “Analysis of grating-assisted backward coupling employing the unified coupled-mode formalism,” J. Opt. Soc. Am. A 16(6), 1303–1311 (1999). [CrossRef]
  28. S. François, S. Fouchet, N. Bouadma, A. Ougazzaden, M. Carré, G. Hervé-Gruyer, M. Filoche, and A. Carenco, “Polarization independent filtering in a grating assisted horizontal directional coupler,” IEEE Photon. Technol. Lett. 7(7), 780–782 (1995). [CrossRef]
  29. Y. Shibata, T. Tamamura, S. Oku, and Y. Kondo, “Coupling coefficient modulation of waveguide grating using sampled grating,” IEEE Photon. Technol. Lett. 6(10), 1222–1224 (1994). [CrossRef]
  30. Q. Guo and W.-P. Huang, “Polarisation-independent optical filters based on co-directional phase-shifted grating-assisted couplers: theory and design,” IEEE Proc. Optoelectron. 143(3), 173–177 (1996). [CrossRef]

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