## Dual transmission band Bragg grating assisted asymmetric directional couplers |

Optics Express, Vol. 19, Issue 2, pp. 1246-1259 (2011)

http://dx.doi.org/10.1364/OE.19.001246

Acrobat PDF (1128 KB)

### 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

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]

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]

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]

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]

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]

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]

**18**(22), 23183–23195 (2010). [CrossRef] [PubMed]

## 2. Bragg grating assisted co-directional phase matching

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. 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]

*et al.*[22

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

*k*=2

*π*/λ, of the grating period Λ, of the BG coupling coefficient

*χ,*and of the waveguide effective index

*n*:

_{eff}*β*

*=*

_{Eq. (1)}*β*

*. From Eq. (1), it follows that:*

_{Eq. (2)}_{1}=Λ

_{2}=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:where

*n*,

_{g1}*n*are the waveguides group indexes and

_{g2}*k*=2

_{Br1}*π*/

*λ*,

_{Br1}*k*=2

_{Br2}*π/λ*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

_{Br2}*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

*n*is smaller or larger than

_{g1}*n*. Each case has then to be separately treated. In what follows, we can arbitrarily assume that

_{g2}*χ*

_{1}≠0 and

*χ*

_{2}= 0.

### 2.1 Bragg grating on the waveguide with higher group index (n_{g1}>n_{g2})

*ω = ω*(

*β*), calculated for

_{eq}*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}.

_{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.

*k>π/*Λ). The red bold line in Fig. 2(a) corresponds to the threshold value

_{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

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

*n*). Figure 3(a) corresponds to the situation where ω

_{g1}<n_{g2}_{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).

*χ*

_{1}values:

*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.

_{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

*χ*

_{1}for co-directional phase matching (

*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

*n*

_{g1}>

*n*

_{g2}or

*n*

_{g1}<

*n*

_{g2}:

*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.

*λ*is explicitly included:

_{ADC}*χ*

_{th}for co-directional phase matching increases with the separation between

*λ*and the BG wavelength

_{ADC}*λ*

_{Br}_{1}. Conversely,

*χ*

_{th}decreases when

*λ*

_{Br}_{1}gets closer to

*λ*, and even cancels when

_{ADC}*λ*

_{Br}_{1}=

*λ*. 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

_{ADC}*λ*. The equality

_{ADC}*λ*

_{Br}_{1}=

*λ*directly leads to

_{ADC}*λ*

_{Br}_{2}=

*λ*

_{Br}_{1}=

*λ*(

_{ADC}*k*=

_{ADC}*k*

_{Br}_{2}=

*k*

_{Br}_{1}) and then

*χ*

_{th}=0.

*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

*disappears as soon as a gap is created around ω*

_{ADC}

_{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.

*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):

*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*) in a similar way as

_{ADC}*χ*

_{th}. The value of

*χ*

_{d}increases with the spectral separation. It cancels for

*λ*

_{Br}_{1}=

*λ*since

_{ADC}*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*). It is uniquely determined by the group indexes of the waveguides that form the ADC:

_{ADC}## 3. Coupled mode approach modeling

_{1}to N

_{5}) 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}=2

*πn*

_{1}/

*λ*, which is introduced in the system of coupled mode equations.

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. 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]

*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

**18**(22), 23183–23195 (2010). [CrossRef] [PubMed]

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

*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).

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

_{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*=–0.5

*µm*^{–1}; ∂

*n*

_{eff}_{2}/∂

*λ*=–0.25

*µm*^{–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. 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]

*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.

*χ*

_{1}=0.5

*χ*

_{th}), there is only one transmission band with Δ

*λ*

_{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}=3

*χ*

_{th}), a narrow peak of Δ

*λ*

_{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

**18**(22), 23183–23195 (2010). [CrossRef] [PubMed]

_{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}.

*χ*

_{1}is increased far above threshold (

*χ*

_{1}=5

*χ*

_{th},

*χ*

_{1}=10

*χ*

_{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) ).

_{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})

_{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*=–0.25

*µm*^{–1}; ∂

*n*

_{eff}_{2}/∂

*λ*=–0.5

*µm*^{–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 3

*χ*

_{th}where

*χ*

_{th}=0.022µm

^{−1}. The example χ

_{1}=2.3

*χ*

_{th}corresponds to the situation of tangential contact between dispersion curves. The grating length is L=1mm.

*χ*

_{1}=0.5

*χ*

_{th}), there is only one transmission band (Δ

*λ*

_{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}=1.5

*χ*

_{th}), in agreement with the phase matching analysis of Section 2, a narrow (Δ

*λ*

_{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.

*χ*

_{1}

*=χ*

_{d}, the two bands merge into a single transmission band whose width (Δ

*λ*

_{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

**18**(22), 23183–23195 (2010). [CrossRef] [PubMed]

*χ*

_{1}and the waveguide effective index modulation Δ

*n*

_{eff}_{1}is:

*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.

_{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.

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

_{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})

*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*=

^{–}0.5

*µm*^{–1}; ∂

*n*

_{eff}_{2}/∂

*λ*=

^{–}0.25

*µm*^{–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.

*χ*

_{1}=0.5

*χ*

_{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}=5

*χ*

_{ref},

*χ*

_{1}=10

*χ*

_{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%.

_{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})

*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*=–0.25

*µm*^{–1}; ∂

*n*

_{eff}_{2}/∂=–0.5

*µm*^{–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 4

*χ*

_{ref}. The grating length is set to be L=0.5mm.

*χ*

_{1}=0.5

*χ*

_{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}=0.5

*χ*

_{ref}) and a moderate grating length, the grating strength (i.e. the product (

*χ*

_{1}L)) 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}=2

*χ*

_{ref},

*χ*

_{1}=4

*χ*

_{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

**18**(22), 23183–23195 (2010). [CrossRef] [PubMed]

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. 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]

**18**(22), 23183–23195 (2010). [CrossRef] [PubMed]

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

## Appendix B: Threshold condition in Eq. (4) and upper limit of *χ*_{1} in Eq(6)

*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.

*χ*

_{1}coefficient (Eq. (4)) is derived in a straightforward manner from Eq. (B3) with the help of Eq. (A2).

*λ*=

*λ*

_{ADC}the effective indexes of both waveguides are equal, the following relation is easily established:

*n*

_{eff}_{1}(λ

_{Br1}) and

*n*

_{eff}_{12}(λ

_{Br1}) 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 |

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. | 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. |

4. | S.-K. Han, R. V. Ramaswamy, and R. F. Tavlykaev, “Highly asymmetrical vertical coupler wavelength filter in InGaAlAs/InP,” Electron. Lett. |

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. |

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. |

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. |

8. | P. Yeh and H. F. Taylor, “Contradirectional frequency-selective couplers for guided-wave optics,” Appl. Opt. |

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. |

10. | R. R. A. Syms, “Optical directional coupler with a grating overlay,” Appl. Opt. |

11. | R. März and H. P. Nolting, “Spectral properties of asymmetrical optical directional couplers with periodic structures,” Opt. Quantum Electron. |

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. |

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. |

14. | T. Erdogan, “Optical add–drop multiplexer based on an asymmetric Bragg coupler,” Opt. Commun. |

15. | I. Baumann, J. Seifert, W. Novak, and M. Sauer, “Compact all-fiber add-drop-multiplexer using fiber Bragg gratings,” IEEE Photon. Technol. Lett. |

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. |

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. | A. Yesayan and R. Vallée, “Optimized grating-frustrated coupler,” Opt. Lett. |

19. | A. Yesayan and R. Vallée, “Zero backreflection condition for a grating-frustrated coupler,” J. Opt. Soc. Am. B |

20. | N. Imoto, “An analysis for contradirectional-coupler-type optical grating filters,” J. Lightwave Technol. |

21. | J. E. Sipe, L. Poladian, and C. M. de Sterke, “Propagation through nonuniform grating structures,” J. Opt. Soc. Am. A |

22. | A. Yariv and M. Nakamura, “Periodic structures for integrated optics,” IEEE J. Quantum Electron. |

23. | R. März and H. P. Nolting, “Spectral properties of asymmetrical optical directional couplers with periodic structures,” Opt. Quantum Electron. |

24. | R. R. A. Syms, “Improved coupled mode theory for codirectionally and contradirectionally coupled waveguide arrays,” J. Opt. Soc. Am. A |

25. | J. Hong and W. P. Huang, “Contra-directional coupling in grating-assisted devices,” J. Lightwave Technol. |

26. | A. Hardy, “A unified approach to coupled-mode phenomena,” IEEE J. Quantum Electron. |

27. | N. Izhaky and A. Hardy, “Analysis of grating-assisted backward coupling employing the unified coupled-mode formalism,” J. Opt. Soc. Am. A |

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. |

29. | Y. Shibata, T. Tamamura, S. Oku, and Y. Kondo, “Coupling coefficient modulation of waveguide grating using sampled grating,” IEEE Photon. Technol. Lett. |

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. |

**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

- 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]
- 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]
- 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).
- 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).
- 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]
- 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]
- 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]
- P. Yeh and H. F. Taylor, “Contradirectional frequency-selective couplers for guided-wave optics,” Appl. Opt. 19(16), 2848–2855 (1980). [CrossRef] [PubMed]
- 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]
- R. R. A. Syms, “Optical directional coupler with a grating overlay,” Appl. Opt. 24(5), 717–726 (1985). [CrossRef] [PubMed]
- 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]
- 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]
- 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]
- T. Erdogan, “Optical add–drop multiplexer based on an asymmetric Bragg coupler,” Opt. Commun. 157(1-6), 249–264 (1998). [CrossRef]
- 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]
- 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]
- 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]
- A. Yesayan and R. Vallée, “Optimized grating-frustrated coupler,” Opt. Lett. 26(17), 1329–1331 (2001). [CrossRef]
- A. Yesayan and R. Vallée, “Zero backreflection condition for a grating-frustrated coupler,” J. Opt. Soc. Am. B 20(7), 1418–1426 (2003). [CrossRef]
- N. Imoto, “An analysis for contradirectional-coupler-type optical grating filters,” J. Lightwave Technol. 3(4), 895–900 (1985). [CrossRef]
- 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]
- A. Yariv and M. Nakamura, “Periodic structures for integrated optics,” IEEE J. Quantum Electron. 13(4), 233–253 (1977). [CrossRef]
- 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]
- 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]
- J. Hong and W. P. Huang, “Contra-directional coupling in grating-assisted devices,” J. Lightwave Technol. 10(7), 873–881 (1992). [CrossRef]
- A. Hardy, “A unified approach to coupled-mode phenomena,” IEEE J. Quantum Electron. 34(7), 1109–1116 (1998). [CrossRef]
- 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]
- 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]
- 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]
- 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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