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This paper addresses the design of narrow band transmission co-directional couplers suitable for wavelength division multiplexing applications. The originality of the proposed asymmetric two-waveguide configuration stems from the use of Bragg gratings operated near band gap to simultaneously achieve high wavelength dispersion and selectivity as well as co-directional phase matching between guides which would be mismatched otherwise. Our theoretical analysis reveals the existence of a minimum Bragg grating coupling strength for co-directional phase matching. The threshold condition is analytically determined, and a coupled mode theory (CMT) four-wave model is successfully applied to describe the behavior of the investigated device. A numerical validation of CMT results is reported in the case of slab waveguides with Bragg grating assisted coupling. The proposed design is shown to be compatible with existing micro-nano-fabrication technology.
©2010 Optical Society of America
1. Introduction
Since their first proposal in the early seventies [1,2], asymmetric directional couplers (ADC) have been the subject of a considerable amount of theoretical [3–5] and experimental investigations [6–12]. Interest in these devices essentially stems from their applications to wavelength selective add-drop demultiplexers [9–12] as well as to wavelength adjustable filter elements in integrated optical circuits or tunable waveguide laser diodes [6–8]. Despite some unique properties of ADCs including a wide wavelength tuning range of more than 100nm [8], the ability for tailoring their spectral response by adjusting the coupling profile [12–15] and the potentially lossless power transfer [3–5], their use is still essentially limited to coarse wavelength division multiplexing (WDM) applications. The reason for this is that their bandwidth is rather large, typically ranging from few nm up to several tens of nm [6,9–15]. The operation compatible with C band WDM applications with 100GHz inter-channel spacing requires however a much narrower bandwidth of approximately 0.5nm.
The 3dB bandwidth of an ADC centered at the wavelength λ0 with phase-matching condition is in the inverse ratio of the coupling length L and of the differential dispersion between the effectives indexes n i of the two waveguide modes [14]:
The standard strategy to enhance the wavelength selectivity consists in increasing the differential dispersion by using highly asymmetric waveguides with either different geometry or different material composition [16–18]. A 0.8 nm bandwidth was thus demonstrated for an ADC with vertical coupling using different InGaAsP quaternary compounds for the guide core layers [16]. Nevertheless, such an asymmetry is not strong enough for making compact devices. The coupling length in the aforementioned realization was approximately 5mm. Strengthening the asymmetry to obtain higher wavelength selectivity and make shorter devices is actually limited by the level of waveguide dispersion achievable with existing materials.
An alternative way to enhance the wavelength selectivity is to take benefit of the strong dispersion of periodically modulated refractive index media. Such a modulation can be implemented either in a direction transverse to the propagation direction of light, using Bragg reflection waveguides (BRW) [19–21], or along the propagation direction using Bragg gratings (BGs) [22,23]. A very narrow bandwidth of 0.29nm with a coupling length of only 1.73mm was demonstrated for an ADC with BRW [21]. However, device operation was based on the coupling between the fundamental mode of a silica waveguide and the first high-order mode of the BRW core. High-order mode operation represents a drawback for telecommunication applications which essentially use single-mode components. Furthermore, BRWs require a non-standard fabrication technology, while conventional planar waveguide fabrication processes are much more desirable for applications.
An ADC coupler with two asymmetric waveguides and a Bragg grating distributed along one waveguide is free from the aforementioned shortcomings. One interesting perspective in this scheme originally proposed by Ankiewicz et al [22,23] is the possibility of gaining in coupler selectivity by using the enhanced dispersion of the BG operated near band gap. However, a careful examination of the phase-matching condition, i.e. the equality of propagation constants in both waveguides, is still to be made for practical applications of such a device.
In this paper, our approach aims at exploiting BG dispersion properties for high performance ADCs. As a major result, we show that BGs can be used not only for achieving high dispersion but also for creating the phase matching condition with an efficient power transfer from the input to drop port (see Fig. 1 ). The scope of this paper is thus to describe fundamental aspects of Bragg grating asymmetric directional couplers (BGADC) and provide basic guiding rules for their design and operation. The paper is organized as follows. Phase-matching conditions for BG-induced co-directional coupling are analyzed in detail in the next section. Results from this analysis are then verified by means of a coupled mode theory (CMT) based four-wave model. Results from the CMT four-wave model are themselves validated by using the CAMFR (Cavity Modeling Framework) numerical software [24,25] in the case of two slab waveguides with BG assisted coupling. This is followed by a summary and concluding remarks.
2. Bragg grating assisted co-directional coupling phase matching
Figure 1 displays a schematic view of the investigated device. It consists of an ADC with BGs. Although this device presents similarities with those previously considered in a number of optical add-drop multiplexers (OADM) involving two coupled waveguides with BG assisted coupling [26–37], its mode of operation is basically different. It neither relies on (i) Bragg exchange contra-directional coupling [24-27 nor on (ii) co-directional coupling mediated Bragg reflection [30–33], nor on (iii) co-directional coupling between two synchronous waveguides frustrated by BG reflection [34–37]. Unlike previous devices which used BG selective reflection, our device is operated out of the BG band gap, and its selectivity results from BG dispersion properties.
The starting point of our analysis is a careful examination of phase-matching conditions in a system of two coupled waveguides where the coupling is assisted by BGs. Previous analyses reported in the literature on this subject are rather incomplete. Either they were restricted to the case of contra-directional coupling [38] or they considered co-directional coupling but ignored the possibility of phase-matching induced by the BG itself [22,23]. The last point is especially important. Indeed, the foundation-stone of our approach is that the BG induced variation of the effective indexes of guided modes can lead to the possibility of co-directional phase matching between two waveguides which would be mismatched otherwise.
To perform the analysis of phase-matching conditions in the asymmetric coupler with Bragg gratings of Fig. 1, we use the effective medium approach [39,40]. This consists in considering BGs as uniform homogeneous media, but with the same dispersion properties as those of the gratings. Assuming a cosine profile of the waveguide corrugation for the sake of simplicity, the relation given by Yariv et al. [41] shows that the equivalent propagation constant 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:
Let us notice that the equivalent index variation described by such a relation is consistent with that established for 1D photonic crystal structures using the more general Floquet-Bloch wave formalism [42].
In what follows, we will assume for the sake of simplicity that the waveguide effective indexes n 1 and n 2 are wavelength independent. Such a crude approximation does not alter the essential of the device physics while it greatly facilitates an insight into its optical properties. Moreover, since the coupling phenomenon occurs in a narrow spectral range, one can also neglect the wavelength dependence of the BG coupling coefficients χ 1 and χ 2. All these parameters will be set constant in the rest of the paper.
By definition, phase-matching means that the propagation constants of the two waveguides are identical for a certain wave-vector, i.e. that β Eq. (1) = β Eq. (2). From Eq. (2) it follows that:
Solutions of Eq. (3) are roots of a fourth-order polynomial. However, their exact analytical expressions are somewhat cumbersome and do not provide a clear answer to the question of interest: how to achieve co-directional phase matching ? Yet, Eq. (3) can be greatly simplified when either both gratings have the same period Λ1 = Λ2 or there is only one BG. It is worthwhile noticing that the case of two BGs with different periods and non-overlapping band gaps can be approximately treated as two distinct sub-cases with only one grating at a time (Λ2 = ∞, χ 2 = 0 and Λ1 = ∞, χ 1 = 0, respectively) provided that the dispersion variation induced by one grating is negligible in the spectral vicinity of the second grating band gap and vice-versa.
Henceforth, we assume that both BGs have the same period Λ. Then, wave-vector solutions of Eq. (3) are:
As seen, these solutions depend on the difference of the squares of the coupling coefficients , but not on χ 1 and χ 2 separately. Therefore, at this stage, the analysis of phase-matching conditions can be reduced to the case where there is only one BG. A further inspection of Eq. (4) shows that the solutions differ according as n 1<n 2 or n 1>n 2. Each case has then to be separately analyzed.
Let us arbitrarily assume χ 1≠0 and χ 2 = 0. The dispersion characteristics, ω = ω(βeq), obtained by means of Eq. (2), are displayed in Figs. 2(a) and 2(b) for several values of χ 1 and the two cases n 1>n 2 and n 1<n 2, respectively. The difference between effective indexes (for instance, n 1 = 3, n 2 = 3.6 in Fig. 2b) and then the values of χ 1 are voluntarily exaggerated to better illustrate the evolution of the dispersion characteristics. The black straight line corresponds to the dispersion characteristic of the uniform waveguide without grating (χ 2 = 0). The family of hyperbolas 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. The BG band gap frequency is centered at ωBr = πc/(Λn 1). The width of the forbidden band is 2cχ 1/n 1.
Fig. 2 In color: dispersion curves of the constant index (n 1) BG waveguide for increasing values of the coupling coefficient χ 1 (from blue to light green). In black: dispersion line of the uniform waveguide with constant index n 2. (a) n 1>n 2 ; (b) n 1<n 2.
A first inspection of dispersion curves in Fig. 2(a) shows that for small values of the coupling coefficient χ 1, the dispersion line of the uniform waveguide only intersects the dispersion curves of the BG waveguide in the region of backward propagating waves. This indicates a contra-directional phase-matching via the Bragg exchange coupling mechanism.
As the coupling coefficient increases, the curvature of the dispersion characteristic of the BG waveguide also increases in the vicinity of the band gap. For a sufficiently high value of the coupling coefficient, the dispersion line of the uniform waveguide can then intersect the BG waveguide dispersion characteristic in the region of forward waves. The red bold line in Fig. 2(a) corresponds to the threshold value 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 . The dispersion characteristics of the two guides intersect at frequencies above the BG band gap.
Clearly, the achievement of co-directional phase-matching excludes the possibility of Bragg exchange contra-directional coupling in the same time: only one mechanism can exist at a time. Neither this mutual exclusion of the two phase-matching mechanisms, nor the existence of a minimum coupling strength for co-directional coupling have been reported in previous studies [22,23,26–38].
In the case where n 1<n 2, the situation is mostly similar. The corresponding dispersion characteristics are shown in Fig. 2(b). Co-directional phase-matching is achieved when the BG coupling coefficient χ 1 is higher than a certain threshold value . One difference with respect to the previous case (n 1>n 2) is that in the region of co-propagative waves, the dispersion characteristics of the two guides intersect at frequencies smaller than the band gap frequency. Another important difference is that the co-directional phase-matching condition is fulfilled only within a limited range of BG coupling coefficients: . The upper limit, χ d, for co-directional phase matching corresponds to a tangential contact between the dispersion characteristics of the two guides. Thus, for χ 1>χ d the condition for co-directional phase-matching is no longer satisfied. It is worthwhile mentioning that contra-directional phase-matching is also excluded in this case.
Previous analysis can be repeated for the situation where χ 1≠0 and χ 2≠0 with the assumption that: χ 1≥χ 2>0. The dispersion diagrams (not shown here) are obviously more complex than those of Figs. 2(a) and 2(b) since each of them is comprised of two series of dispersion curves instead of one. However, the general conclusions are unchanged. There is a minimum value of χ 1 (χ 1 = χ th) for co-directional coupling whether n 1 is larger or smaller than n 2. There is an upper limit χ 1 = χ d for co-directional coupling in the case where n 1<n 2. Expressions of χ th and χ d can be derived from Eqs. (2)–(4) using our simple model with constant index waveguides:
Details on formula derivation can be found in Appendix A. As seen, a unique threshold formula can be used for the two cases n 1>n 2 and n 1<n 2. Equation (5) also shows that χ 2≠0 and then the presence of a second BG increase 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 a BG in only one of the two ADC waveguides. The hypothesis χ 2 = 0 will then be kept for the rest of the paper.
3. Coupled mode approach modeling
A coupled mode theory approach is used to verify the assertions inferred from the phase-matching analysis of Section 2 and to explore the BGADC behavior in detail. The investigated device is schematically represented in Fig. 3 . It consists of two parallel slab waveguides surrounded by a cladding. The waveguide in the upper part bears a double side BG of p periods with total length L = pΛ. The surrounding cladding index is denoted Nb, and the refractive indices of waveguides 1 and 2 are N1 and N2, respectively. Both waveguides are assumed to fulfill the single mode operation condition. The dashed line traced for the BG assisted waveguide indicates the original waveguide width without grating modulation. This width is used to determine the effective index and the corresponding propagation constant , 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 [43–46], 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 the individual waveguides modes [29,45,46]. The relation governing the exchange between forward and backward propagating waves of two coupled waveguides on the ± grating sections is:
where A is the column vector A = [a1 +, a1 -, a2 +, a2 - ]T with field amplitude elements and M+, M - are 4 × 4 matrices with constant value coefficients:The waveguide propagation constants β 1, β 2 and the BG coupling coefficients χ 1 and χ 2 appearing in Eq. (8) have already been defined in Section 2. The newly appearing terms κ and χ 12 stand for the co-directional evanescent-coupling and contra-directional Bragg exchange evanescent-coupling, respectively. Using a standard diagonalization procedure, the system of linear differential Eqs. (7) and (8) can be reduced to the equations for the normal modes:
where B = [b1 +, b1 -, b2 +, b2 - ]T is the new column vector and Γ ± , the eigenvalue diagonal matrix:Here γ 1 + and γ 2 ± are the eigenvalues of M± . In other words, they identify with the propagation constants of normal modes in the parallel coupler.
By direct integration, the transfer matrix for the normal modes in each uniform section is:
where:and Λ ± is the length of the ± section. The transfer matrix T1 for one period Λ = Λ+ + Λ- is then:where O ± is the eigenvectors matrix of M± in the ± section. The transfer matrix for the whole structure is then:The known boundary conditions are the amplitudes of the forward modes a1 +(0), a2 +(0) at the device entrance and the amplitudes of the backward modes a1 -(L), a2 -(L) at the device end. The purpose is to find the reflected modes a1 -(0), a2 −0) and the transmitted modes a1 +(L), a2 +(L). The mode amplitudes at the device entrance and the device end are related to each other in the more explicit form:
To avoid undesirable reflection for optimal device operation, the incident light has to be injected through the uniform waveguide 2 (Fig. 3). Denoting R|| -the thru-port direct reflection- and R × -the drop-port exchange reflection-, the column vector associated to mode amplitudes at the device input writes:
The vector of mode amplitudes at the device output end is then:
where T ||, T × are the thru-port direct transmission and the drop-port exchange transmission, respectively. The system of linear Eq. (15) can be solved by elementary algebra. Final expressions for the transmission and reflection coefficients T//, Tx, R //, R x are given in Appendix B.In order to numerically exploit the CMT model, coefficients κ and χ 12 entering in the definition of matrix M± Eq. (8) need to be expressed versus the geometrical parameters of the coupler. A conventional expression corresponding to a complete crossover can actually be used for the co-directional evanescent coupling coefficient κ:
In what follows, we will consider situations where κ is much smaller than the threshold coupling coefficient for co-directional coupling χ th. On the other hand, the contra-directional Bragg exchange evanescent-coupling coefficient χ 12 is approximated as follows:
Arguments justifying this approximation can be found in Appendix C. Since χ 1Λ<<1 in the following examples, Eq. (19) shows that χ 12<<κ.
Our coupled mode theory approach is then applicable to the numerical modeling of the BGADC behavior in any condition. Of particular interest is the comparison between its spectral responses below and above the threshold for co-directional coupling as previously defined in the phase matching analysis of Section 2. Let us notice, however, that the threshold value given in Eq. (5) must now be multiplied by a factor of π/2 to account for the rectangular shape of the grating (Fig. 3) [14]. Figures 4(a) and 4(b) show the spectral dependence of the co-directional coupling transmission T × for several values of χ 1 (with χ 2 = 0) and for the two cases n 1>n 2 and n 1<n 2, respectively.
Fig. 4 Drop-port exchange transmission spectral response for several values of the coupling coefficient χ 1. (a) n 1 = 3.255, n 2 = 3.25 (b) n 1 = 3.255, n 2 = 3.26.
For the first case (n 1>n 2), we take n 1 = 3.255 and n 2 = 3.25. The grating period is chosen to be Λ = 0.238µm to fix the Bragg wavelength at 1.55µm for a 50% grating duty ratio. The total grating length is 1000µm. The different values of χ 1 are: 0.5, 3, 12, and 30 χ th, with χ th = 0.032µm−1. The transmission spectra of Fig. 4(a) confirm that below threshold, the drop-port exchange transmission T × is low. Its amplitude is indeed of the same order as the amplitudes of secondary peaks. Just above threshold (χ 1 = 3χ th), a narrow transmission peak appears at a frequency slightly larger than the Bragg frequency, but the transmission does not reach its maximum. 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. This will be illustrated later in Fig. 5 . As χ 1 is increased far above threshold (χ 1 = 12χ th, χ 1 = 30χ th) the transmission peak progressively broadens and shifts to higher frequencies. The peak amplitude is rapidly growing to its asymptotic limit while the phase matching frequency moves away from the Bragg frequency [see Fig. 2(a)].
Fig. 5 Transmission and reflection spectra for thru-and drop-ports. (a) n 1>n 2 (b) n 1<n 2. Insets show the enlarged view in the vicinity of the phase matching wavelengths.
For the second case (n 1<n 2), we keep n 1 = 3.255 and set n 2 = 3.26. The other parameters are unchanged. As seen in Fig. 4(b), below threshold, the transmission T × is low. Just above threshold, a narrow transmission peak appears at a frequency slightly smaller than the gap frequency. As χ 1 is increased, the peak broadens and now shifts to the lower frequency side of the band gap. A still bigger difference with previous behavior of Fig. 4(a) occurs for very high values of χ 1 (here χ 1 = 30χ th). Indeed, above the limit χ 1 = χ d, the phase matching condition is no more satisfied and the transmission T × is low again. One may admit, however, that χ d is rather high and hardly attainable in practice. For practical applications, a good compromise between a high selectivity and a high transmission level will be to choose χ 1 not too high, just a few times the threshold value.
4. CAMFR numerical modeling
The numerical solver CAMFR based on an eigenmode propagation method is used to validate the results of the CMT four-wave model. Numerical modeling is performed in the case of two slab waveguides of Fig. 3 with one Bragg grating. For each waveguide, we assume the same N1 = N2 = 3.32 core index material, embedded in a background cladding material with refractive index Nb = 3.17. As for the for CMT four-wave model, the Bragg wavelength is fixed at 1.55µm. The grating period is Λ = 0.238µm for a 50% grating duty ratio. The total grating length is 1000µm. For a rectangular grating profile, the relation between the BG coupling coefficient χ 1 and the waveguide effective index modulation Δn 1 is:
We set χ 1 = 3.14χ th. This imposes a grating profile with a minimal waveguide width of 0.41µm and a maximal waveguide width of 0.93µm. The width of the second waveguide is adjusted so as to explore each of the cases n 1>n 2 and n 1<n 2. For n 1>n 2, the width of the second waveguide is 0.53µm. The mean separation distance between the waveguides is 1.7µm. The light is injected through the second waveguide without BG.
The spectral responses for thru-port and drop-port transmissions and reflections are represented in Fig. 5(a). Numerical modeling results confirm the presence of a narrow (1.1nm half-width) transmission peak T × as in Fig. 4(a) for χ 1 = 3χ th. Co-directional coupling to the Bragg waveguide is achieved with around 80% transmitted power at maximum. The remaining fraction of power (20%) is essentially reflected into the input waveguide via combined co-directional coupling and Bragg reflection mechanisms. The drop-port reflection R x is low as well as the residual thru-port transmission T //. The selectivity obtained for a 1mm device length is approximately a factor of four higher than that in conventional approaches [16–18]. It is worthwhile noticing that the excess loss due to reflection does not depend on the device length, provided that the dispersion variation induced by the BG reaches its limit. The 100 GHz (0.5 nm) inter-channel spacing or bandwidth required for C band WDM applications can then be simply achieved by increasing the device length by a factor of two without additional loss penalty.
For n 1<n 2, the width of the second waveguide is 0.67µm while the BG parameters are let unchanged. The mean separation distance between the waveguides is 1.6µm. Figure 5(b) shows the transmission and reflection spectra. Results are similar to those of Fig. 5(a), except for the fact that the phase matching wavelength is now located on the long wavelength side of the band gap.
The CAMFR results then perfectly agree with those of the CMT four-wave model. In other words, the essential of the device physics is described by CMT. It merits to be noted that the effective indexes of slab waveguides chosen in Figs. 4 and 5 are quite similar to those of refractive index waveguides in the InGaAsP/InP system. All the waveguide dimensions are also compatible with existing micro-nano-fabrication technology.
5. Summary and conclusions
The aim of our work was to propose a new design for co-directional wavelength selective couplers whose selectivity is no longer limited by the dispersive properties of refractive waveguides made of existing materials, bur rather controlled by an artificial dispersion obtained by material structuring.
Our approach has made use of the dispersive properties of a Bragg grating (BG) in the vicinity of its own photonic band gap. It has been shown that the change of the waveguide dispersion induced by the BG could be successfully used to achieve phase matching conditions is a system of otherwise mismatched waveguides. This point marks an important difference with previous works performed in this domain [22,23].
The phase matching analysis has revealed the existence of a threshold condition for the BG coupling coefficient. The use of a simplified model with constant effective index waveguides has provided an analytical expression for this threshold as well as a general description of the coupler behavior. It has been shown that the optimal operation was achieved with only one Bragg grating distributed along one of the two waveguides. Different device behaviors were predicted according as the grating was placed along the high- or low- index waveguide.
It should be noticed that the threshold coupling condition is an essentially new aspect of Bragg grating asymmetric directional couplers (BGADC) as compared to conventional co-directional couplers and Bragg reflectors. Indeed, a complete power transfer can be achieved with conventional co-directional couplers whatever the coupling coefficient may be. It suffices that the device length satisfies the condition expressed by Eq. (18). In a similar way, Bragg reflectors can exhibit extremely high reflection levels with arbitrarily low BG coupling coefficient χ. The only condition is that the grating coupling strength be sufficiently high: χL>>1. Because of their specificity, BGADCs then exhibit new properties that do not simply result from the individual properties of their constitutive elements.
Theoretical predictions from the analytical model have been successfully verified using a coupled mode theory (CMT) four-wave model, which has been developed in this work. CMT results have been validated in turn using the free available software CAMFR based on an eigenmode propagation method. The validation has been carried out in the case of a coupler formed by two slab waveguides with one grating implemented in one of the two waveguides. The excellent agreement between the two models indicates that the essential of the device physics can be described by CMT. Results from numerical modeling have shown the possibility of obtaining a 1.1nm bandwidth at telecommunication wavelengths with a 1mm long device and only 1dB of reflection loss penalty. This selectivity is a factor of four higher than that previously reported for conventional approaches. The proposed design is compatible with existing fabrication technology.
Our results here validated by 2D numerical methods can be obviously extended to 3D structures. The coupled mode approach accounts for the essential of the device physics and applies to any structure whether it is described by a 2D or 3D geometry.
Present results could be further improved by considering designs with non-uniform gratings. It can also be extended to other types of dispersive elements such as photonic crystal waveguides.
Appendix A. Threshold condition in Eq. (5) and upper limit of χ1 in Eq. (6)
For the sake of generality, let us assume that each waveguide includes a Bragg grating with period Λ. As can be seen from Fig. 2, the threshold for co-directional coupling is reached when the real parts of the propagation constants are both equal to π/Λ. The corresponding frequency is larger than the Bragg frequency when n 1>n 2. The reverse situation occurs for n 1<n 2. The wave-vector meeting the phase matching condition can then be directly determined from Eq. (2):
The sign + (or −) in Eq. (A1) depends on whether n 1>n 2 or n 1<n 2. Equation (5) in Section 2 is then obtained in a straightforward manner from Eq. (A1).
Concerning the derivation of Eq. (6), it is worthwhile noticing that the upper limit for co-directional coupling χ 1=χ d corresponds to a tangential contact between the dispersion curves of the two guides. In other words, there is only one solution for Eq. (4). Equation (6) in Section 2 is then readily obtained by elementary algebra.
Appendix B. Transmission and reflection coefficients T//, Tx, R//, Rx
The system of linear Eq. (15) is solved by elementary algebra. Final expressions for the reflection and transmission coefficients T//, Tx, R //, R x are:
Appendix C. Bragg exchange evanescent coupling coefficient in Eq. (19)
The Bragg exchange evanescent coupling coefficient χ 12 is usually determined from the overlap integral between the forward propagating mode of one waveguide and the backward propagating mode of the other waveguide in the region where the index is modified by the Bragg grating. On the other hand, the co-directional coupling coefficient κ is determined from the overlap between the mode propagating in one waveguide with the tail of the mode propagating in the adjacent waveguide. A fraction of this overlap, which corresponds to the to BG-perturbed-index region, is but the Bragg exchange evanescent coupling coefficient itself, χ 12. This fraction, χ 12/κ, can then be approximated by the ratio of the grating index modulation to the waveguide effective index. Assuming that each waveguide includes a BG with period Λ, one can write:
Equation (19) in Section 3 is directly obtained by combining this expression with Eq. (20) in Section 4.
References and links
1. H. F. Taylor, “Frequency-selective coupling in parallel dielectric waveguides,” Opt. Commun. 8(4), 421–425 (1973). [CrossRef]
2. C. Elachi and C. Yeh, “Frequency selective coupler for integrated optics systems,” Opt. Commun. 7(3), 201–204 (1973). [CrossRef]
3. D. Marcuse, “Directional couplers made of nonidentical asymmetric slabs. Part I: Synchronous couplers,” J. Lightwave Technol. 5(1), 113–118 (1987). [CrossRef]
4. D. Marcuse, “Directional couplers made of nonidentical asymmetric slabs. Part II: Grating-assisted couplers,” J. Lightwave Technol. 5(2), 268–273 (1987). [CrossRef]
5. W.-P. Huang, “Coupled mode theory for optical waveguides: an overview,” J. Opt. Soc. Am. A 11(3), 963–983 (1994). [CrossRef]
6. R. C. Alferness and R. V. Schmidt, “Tunable optical waveguide directional coupler filter,” Appl. Phys. Lett. 33(2), 161–163 (1978). [CrossRef]
7. R. C. Alferness, U. Koren, L. L. Buhl, B. I. Miller, M. G. Young, T. L. Koch, G. Raybon, and C. A. Burrus, “Broadly tunable InGaAsP/InP laser based on a vertical coupler filter with 57-nm tuning range,” Appl. Phys. Lett. 60(26), 3209–3211 (1992). [CrossRef]
8. P.-J. Rigole, S. Nilsson, L. Backbom, T. Klinga, J. Wallin, B. Stalnacke, E. Berglind, and B. Stoltz, “114-nm wavelength tuning range of a vertical grating assisted codirectional coupler laser with a super structure grating distributed Bragg reflector,” IEEE Photon. Technol. Lett. 7(7), 697–699 (1995). [CrossRef]
9. C. Bornholdt, 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]
10. Y.-H. Jan, M. E. Heimbuch, L. A. Coldren, and S. P. DenBaars, “InP/InGaAsP grating-assisted codirectional coupler tunable receiver with a 30nm wavelength tuning range,” Electron. Lett. 32(18), 1697–1698 (1996). [CrossRef]
11. 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]
12. A. Lupu, P. Win, H. Sik, P. Boulet, M. Carré, 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]
13. R. C. Alferness, “Optical directional couplers with weighted coupling,” Appl. Phys. Lett. 35, 109–126 (1978).
14. K. A. Winick, “Design of grating-assisted waveguide couplers with weighted coupling,” J. Lightwave Technol. 9(11), 1481–1492 (1991). [CrossRef]
15. B. E. Little, C. Wu, and W. P. Huang, “Synthesis of codirectional couplers with ultralow sidelobes and minimum bandwidth,” Opt. Lett. 20(11), 1259–1261 (1995). [CrossRef] [PubMed]
16. 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]
17. 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).
18. S.-K. Han, R. V. Ramaswamy, and R. F. Tavlykaev, “Highly asymmetrical vertical coupler wavelength filter in InGaAlAs/InP,” Electron. Lett. 33, 31 (1999).
19. B. R. West and A. S. Helmy, “Dispersion tailoring of the quarter-wave Bragg reflection waveguide,” Opt. Express 14(9), 4073–4086 (2006). [CrossRef] [PubMed]
20. R. Das and K. Thyagarajan, “Strong dispersive features of dual Bragg cladding waveguides using higher-order mode-coupling mechanism,” J. Opt. A, Pure Appl. Opt. 11(10), 105410 (2009). [CrossRef]
21. M. Dainese, M. Swillo, L. Wosinski, and L. Thylen, “Directional coupler wavelength selective filter based on dispersive Bragg reflection waveguide,” Opt. Commun. 260(2), 514–521 (2006). [CrossRef]
22. A. Ankiewicz and G.-D. Peng, “Narrow bandpass filter using Bragg grating coupler in transmission mode,” Electron. Lett. 33(25), 2151–2153 (1997). [CrossRef]
23. A. Ankiewicz, Z. H. Wang, and G.-D. Peng, “Analysis of narrow bandpass filter using coupler with Bragg grating in transmission,” Opt. Commun. 156(1-3), 27–31 (1998). [CrossRef]
24. P. Bienstman and R. Baets, “Optical Modelling of Photonic Crystals and VCSELs using Eigenmode Expansion and Perfectly Matched Layers,” Opt. Quantum Electron. 33(4/5), 327–341 (2001). [CrossRef]
25. http://camfr.sourceforge.net.
26. P. Yeh and H. F. Taylor, “Contradirectional frequency-selective couplers for guided-wave optics,” Appl. Opt. 19(16), 2848–2855 (1980). [CrossRef] [PubMed]
27. 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]
28. R. R. A. Syms, “Optical directional coupler with a grating overlay,” Appl. Opt. 24(5), 717–726 (1985). [CrossRef] [PubMed]
29. 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]
30. 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]
31. 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]
32. T. Erdogan, “Optical add–drop multiplexer based on an asymmetric Bragg coupler,” Opt. Commun. 157(1-6), 249–264 (1998). [CrossRef]
33. I. Baumann, J. Seifert, W. Nowak, and M. Sauer, “Compact all-fiber add-drop-multiplexer using fiber Bragg gratings,” IEEE Photon. Technol. Lett. 8(10), 1331–1333 (1996). [CrossRef]
34. 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]
35. 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]
36. A. Yesayan and R. Vallée, “Optimized grating-frustrated coupler,” Opt. Lett. 26(17), 1329–1331 (2001). [CrossRef]
37. A. Yesayan and R. Vallée, “Zero backreflection condition for a grating-frustrated coupler,” J. Opt. Soc. Am. B 20(7), 1418–1426 (2003). [CrossRef]
38. N. Imoto, “An analysis for contradirectional-coupler-type optical grating filters,” J. Lightwave Technol. 3(4), 895–900 (1985). [CrossRef]
39. 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]
40. A. Arraf, L. Poladian, C. M. de Sterke, and T. G. Brown, “Effective-medium approach for counterpropagating waves in nonuniform Bragg gratings,” J. Opt. Soc. Am. A 14(5), 1137–1143 (1997). [CrossRef]
41. A. Yariv and M. Nakamura, “Periodic structures for integrated optics,” IEEE J. Quantum Electron. 13(4), 233–253 (1977). [CrossRef]
42. B. Lombardet, L. A. Dunbar, R. Ferrini, and R. Houdré, “Fourier analysis of Bloch wave propagation in photonic crystals,” J. Opt. Soc. Am. B 22(6), 1179–1190 (2005). [CrossRef]
43. 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]
44. J. Hong and W. P. Huang, “Contra-directional coupling in grating-assisted guided-wave devices,” J. Lightwave Technol. 10(7), 873–881 (1992). [CrossRef]
45. A. Hardy, “A unified approach to coupled-mode phenomena,” IEEE J. Quantum Electron. 34(7), 1109–1116 (1998). [CrossRef]
46. 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]
